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Properties

Label 400.4.a.u.1.1
Level $400$
Weight $4$
Character 400.1
Self dual yes
Analytic conductor $23.601$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,4,Mod(1,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 400.a (trivial)

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: yes
Analytic conductor: \(23.6007640023\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 40)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 400.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+10.0000 q^{3} -18.0000 q^{7} +73.0000 q^{9} +16.0000 q^{11} +6.00000 q^{13} +6.00000 q^{17} +124.000 q^{19} -180.000 q^{21} +42.0000 q^{23} +460.000 q^{27} +142.000 q^{29} +188.000 q^{31} +160.000 q^{33} -202.000 q^{37} +60.0000 q^{39} +54.0000 q^{41} +66.0000 q^{43} +38.0000 q^{47} -19.0000 q^{49} +60.0000 q^{51} -738.000 q^{53} +1240.00 q^{57} -564.000 q^{59} -262.000 q^{61} -1314.00 q^{63} -554.000 q^{67} +420.000 q^{69} -140.000 q^{71} -882.000 q^{73} -288.000 q^{77} +1160.00 q^{79} +2629.00 q^{81} +642.000 q^{83} +1420.00 q^{87} -854.000 q^{89} -108.000 q^{91} +1880.00 q^{93} +478.000 q^{97} +1168.00 q^{99} +O(q^{100})\)

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
\(3\) 10.0000 1.92450 0.962250 − 0.272166i \(-0.0877398\pi\)
0.962250 + 0.272166i \(0.0877398\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −18.0000 −0.971909 −0.485954 − 0.873984i \(-0.661528\pi\)
−0.485954 + 0.873984i \(0.661528\pi\)
\(8\) 0 0
\(9\) 73.0000 2.70370
\(10\) 0 0
\(11\) 16.0000 0.438562 0.219281 − 0.975662i \(-0.429629\pi\)
0.219281 + 0.975662i \(0.429629\pi\)
\(12\) 0 0
\(13\) 6.00000 0.128008 0.0640039 − 0.997950i \(-0.479613\pi\)
0.0640039 + 0.997950i \(0.479613\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 0.0856008 0.0428004 − 0.999084i \(-0.486372\pi\)
0.0428004 + 0.999084i \(0.486372\pi\)
\(18\) 0 0
\(19\) 124.000 1.49724 0.748620 − 0.663000i \(-0.230717\pi\)
0.748620 + 0.663000i \(0.230717\pi\)
\(20\) 0 0
\(21\) −180.000 −1.87044
\(22\) 0 0
\(23\) 42.0000 0.380765 0.190383 − 0.981710i \(-0.439027\pi\)
0.190383 + 0.981710i \(0.439027\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 460.000 3.27878
\(28\) 0 0
\(29\) 142.000 0.909267 0.454633 − 0.890679i \(-0.349770\pi\)
0.454633 + 0.890679i \(0.349770\pi\)
\(30\) 0 0
\(31\) 188.000 1.08922 0.544610 − 0.838690i \(-0.316678\pi\)
0.544610 + 0.838690i \(0.316678\pi\)
\(32\) 0 0
\(33\) 160.000 0.844013
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −202.000 −0.897530 −0.448765 − 0.893650i \(-0.648136\pi\)
−0.448765 + 0.893650i \(0.648136\pi\)
\(38\) 0 0
\(39\) 60.0000 0.246351
\(40\) 0 0
\(41\) 54.0000 0.205692 0.102846 − 0.994697i \(-0.467205\pi\)
0.102846 + 0.994697i \(0.467205\pi\)
\(42\) 0 0
\(43\) 66.0000 0.234068 0.117034 − 0.993128i \(-0.462661\pi\)
0.117034 + 0.993128i \(0.462661\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 38.0000 0.117933 0.0589667 − 0.998260i \(-0.481219\pi\)
0.0589667 + 0.998260i \(0.481219\pi\)
\(48\) 0 0
\(49\) −19.0000 −0.0553936
\(50\) 0 0
\(51\) 60.0000 0.164739
\(52\) 0 0
\(53\) −738.000 −1.91268 −0.956341 − 0.292255i \(-0.905595\pi\)
−0.956341 + 0.292255i \(0.905595\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1240.00 2.88144
\(58\) 0 0
\(59\) −564.000 −1.24452 −0.622259 − 0.782812i \(-0.713785\pi\)
−0.622259 + 0.782812i \(0.713785\pi\)
\(60\) 0 0
\(61\) −262.000 −0.549929 −0.274964 − 0.961454i \(-0.588666\pi\)
−0.274964 + 0.961454i \(0.588666\pi\)
\(62\) 0 0
\(63\) −1314.00 −2.62775
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −554.000 −1.01018 −0.505089 − 0.863067i \(-0.668540\pi\)
−0.505089 + 0.863067i \(0.668540\pi\)
\(68\) 0 0
\(69\) 420.000 0.732783
\(70\) 0 0
\(71\) −140.000 −0.234013 −0.117007 − 0.993131i \(-0.537330\pi\)
−0.117007 + 0.993131i \(0.537330\pi\)
\(72\) 0 0
\(73\) −882.000 −1.41411 −0.707057 − 0.707157i \(-0.749977\pi\)
−0.707057 + 0.707157i \(0.749977\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −288.000 −0.426242
\(78\) 0 0
\(79\) 1160.00 1.65203 0.826014 − 0.563650i \(-0.190603\pi\)
0.826014 + 0.563650i \(0.190603\pi\)
\(80\) 0 0
\(81\) 2629.00 3.60631
\(82\) 0 0
\(83\) 642.000 0.849020 0.424510 − 0.905423i \(-0.360446\pi\)
0.424510 + 0.905423i \(0.360446\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1420.00 1.74988
\(88\) 0 0
\(89\) −854.000 −1.01712 −0.508561 − 0.861026i \(-0.669822\pi\)
−0.508561 + 0.861026i \(0.669822\pi\)
\(90\) 0 0
\(91\) −108.000 −0.124412
\(92\) 0 0
\(93\) 1880.00 2.09620
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 478.000 0.500346 0.250173 − 0.968201i \(-0.419513\pi\)
0.250173 + 0.968201i \(0.419513\pi\)
\(98\) 0 0
\(99\) 1168.00 1.18574
\(100\) 0 0
\(101\) −1794.00 −1.76742 −0.883711 − 0.468033i \(-0.844963\pi\)
−0.883711 + 0.468033i \(0.844963\pi\)
\(102\) 0 0
\(103\) 642.000 0.614157 0.307078 − 0.951684i \(-0.400649\pi\)
0.307078 + 0.951684i \(0.400649\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −850.000 −0.767968 −0.383984 − 0.923340i \(-0.625448\pi\)
−0.383984 + 0.923340i \(0.625448\pi\)
\(108\) 0 0
\(109\) 666.000 0.585241 0.292620 − 0.956229i \(-0.405473\pi\)
0.292620 + 0.956229i \(0.405473\pi\)
\(110\) 0 0
\(111\) −2020.00 −1.72730
\(112\) 0 0
\(113\) 1446.00 1.20379 0.601895 − 0.798575i \(-0.294413\pi\)
0.601895 + 0.798575i \(0.294413\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 438.000 0.346095
\(118\) 0 0
\(119\) −108.000 −0.0831962
\(120\) 0 0
\(121\) −1075.00 −0.807663
\(122\) 0 0
\(123\) 540.000 0.395855
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −1154.00 −0.806307 −0.403153 − 0.915132i \(-0.632086\pi\)
−0.403153 + 0.915132i \(0.632086\pi\)
\(128\) 0 0
\(129\) 660.000 0.450463
\(130\) 0 0
\(131\) 368.000 0.245437 0.122719 − 0.992441i \(-0.460839\pi\)
0.122719 + 0.992441i \(0.460839\pi\)
\(132\) 0 0
\(133\) −2232.00 −1.45518
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 670.000 0.417825 0.208912 − 0.977934i \(-0.433008\pi\)
0.208912 + 0.977934i \(0.433008\pi\)
\(138\) 0 0
\(139\) 572.000 0.349039 0.174519 − 0.984654i \(-0.444163\pi\)
0.174519 + 0.984654i \(0.444163\pi\)
\(140\) 0 0
\(141\) 380.000 0.226963
\(142\) 0 0
\(143\) 96.0000 0.0561393
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −190.000 −0.106605
\(148\) 0 0
\(149\) 1730.00 0.951189 0.475594 − 0.879665i \(-0.342233\pi\)
0.475594 + 0.879665i \(0.342233\pi\)
\(150\) 0 0
\(151\) −1324.00 −0.713547 −0.356773 − 0.934191i \(-0.616123\pi\)
−0.356773 + 0.934191i \(0.616123\pi\)
\(152\) 0 0
\(153\) 438.000 0.231439
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2946.00 −1.49756 −0.748778 − 0.662820i \(-0.769359\pi\)
−0.748778 + 0.662820i \(0.769359\pi\)
\(158\) 0 0
\(159\) −7380.00 −3.68096
\(160\) 0 0
\(161\) −756.000 −0.370069
\(162\) 0 0
\(163\) 2098.00 1.00815 0.504074 − 0.863661i \(-0.331834\pi\)
0.504074 + 0.863661i \(0.331834\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −866.000 −0.401276 −0.200638 − 0.979665i \(-0.564301\pi\)
−0.200638 + 0.979665i \(0.564301\pi\)
\(168\) 0 0
\(169\) −2161.00 −0.983614
\(170\) 0 0
\(171\) 9052.00 4.04809
\(172\) 0 0
\(173\) 1678.00 0.737433 0.368717 − 0.929542i \(-0.379797\pi\)
0.368717 + 0.929542i \(0.379797\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −5640.00 −2.39508
\(178\) 0 0
\(179\) 1620.00 0.676450 0.338225 − 0.941065i \(-0.390174\pi\)
0.338225 + 0.941065i \(0.390174\pi\)
\(180\) 0 0
\(181\) 2510.00 1.03076 0.515378 − 0.856963i \(-0.327652\pi\)
0.515378 + 0.856963i \(0.327652\pi\)
\(182\) 0 0
\(183\) −2620.00 −1.05834
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 96.0000 0.0375413
\(188\) 0 0
\(189\) −8280.00 −3.18667
\(190\) 0 0
\(191\) 372.000 0.140927 0.0704633 − 0.997514i \(-0.477552\pi\)
0.0704633 + 0.997514i \(0.477552\pi\)
\(192\) 0 0
\(193\) −2938.00 −1.09576 −0.547880 − 0.836557i \(-0.684565\pi\)
−0.547880 + 0.836557i \(0.684565\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2234.00 −0.807949 −0.403974 − 0.914770i \(-0.632372\pi\)
−0.403974 + 0.914770i \(0.632372\pi\)
\(198\) 0 0
\(199\) 3048.00 1.08576 0.542882 − 0.839809i \(-0.317333\pi\)
0.542882 + 0.839809i \(0.317333\pi\)
\(200\) 0 0
\(201\) −5540.00 −1.94409
\(202\) 0 0
\(203\) −2556.00 −0.883724
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 3066.00 1.02948
\(208\) 0 0
\(209\) 1984.00 0.656632
\(210\) 0 0
\(211\) −4896.00 −1.59741 −0.798707 − 0.601720i \(-0.794482\pi\)
−0.798707 + 0.601720i \(0.794482\pi\)
\(212\) 0 0
\(213\) −1400.00 −0.450359
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −3384.00 −1.05862
\(218\) 0 0
\(219\) −8820.00 −2.72146
\(220\) 0 0
\(221\) 36.0000 0.0109576
\(222\) 0 0
\(223\) −5302.00 −1.59214 −0.796072 − 0.605202i \(-0.793092\pi\)
−0.796072 + 0.605202i \(0.793092\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −3778.00 −1.10465 −0.552323 − 0.833630i \(-0.686259\pi\)
−0.552323 + 0.833630i \(0.686259\pi\)
\(228\) 0 0
\(229\) −3034.00 −0.875513 −0.437756 − 0.899094i \(-0.644227\pi\)
−0.437756 + 0.899094i \(0.644227\pi\)
\(230\) 0 0
\(231\) −2880.00 −0.820303
\(232\) 0 0
\(233\) 3478.00 0.977903 0.488951 − 0.872311i \(-0.337380\pi\)
0.488951 + 0.872311i \(0.337380\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 11600.0 3.17933
\(238\) 0 0
\(239\) −1560.00 −0.422209 −0.211105 − 0.977463i \(-0.567706\pi\)
−0.211105 + 0.977463i \(0.567706\pi\)
\(240\) 0 0
\(241\) −3218.00 −0.860123 −0.430061 − 0.902800i \(-0.641508\pi\)
−0.430061 + 0.902800i \(0.641508\pi\)
\(242\) 0 0
\(243\) 13870.0 3.66157
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 744.000 0.191658
\(248\) 0 0
\(249\) 6420.00 1.63394
\(250\) 0 0
\(251\) −688.000 −0.173013 −0.0865063 − 0.996251i \(-0.527570\pi\)
−0.0865063 + 0.996251i \(0.527570\pi\)
\(252\) 0 0
\(253\) 672.000 0.166989
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2170.00 −0.526696 −0.263348 − 0.964701i \(-0.584827\pi\)
−0.263348 + 0.964701i \(0.584827\pi\)
\(258\) 0 0
\(259\) 3636.00 0.872317
\(260\) 0 0
\(261\) 10366.0 2.45839
\(262\) 0 0
\(263\) 2274.00 0.533159 0.266580 − 0.963813i \(-0.414106\pi\)
0.266580 + 0.963813i \(0.414106\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −8540.00 −1.95745
\(268\) 0 0
\(269\) 7146.00 1.61970 0.809850 − 0.586637i \(-0.199548\pi\)
0.809850 + 0.586637i \(0.199548\pi\)
\(270\) 0 0
\(271\) −2604.00 −0.583696 −0.291848 − 0.956465i \(-0.594270\pi\)
−0.291848 + 0.956465i \(0.594270\pi\)
\(272\) 0 0
\(273\) −1080.00 −0.239431
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5150.00 1.11709 0.558544 − 0.829475i \(-0.311360\pi\)
0.558544 + 0.829475i \(0.311360\pi\)
\(278\) 0 0
\(279\) 13724.0 2.94493
\(280\) 0 0
\(281\) 5270.00 1.11880 0.559398 − 0.828899i \(-0.311032\pi\)
0.559398 + 0.828899i \(0.311032\pi\)
\(282\) 0 0
\(283\) 3434.00 0.721308 0.360654 − 0.932700i \(-0.382554\pi\)
0.360654 + 0.932700i \(0.382554\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −972.000 −0.199914
\(288\) 0 0
\(289\) −4877.00 −0.992673
\(290\) 0 0
\(291\) 4780.00 0.962916
\(292\) 0 0
\(293\) 9878.00 1.96955 0.984776 − 0.173826i \(-0.0556132\pi\)
0.984776 + 0.173826i \(0.0556132\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 7360.00 1.43795
\(298\) 0 0
\(299\) 252.000 0.0487409
\(300\) 0 0
\(301\) −1188.00 −0.227492
\(302\) 0 0
\(303\) −17940.0 −3.40141
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 8054.00 1.49728 0.748642 − 0.662975i \(-0.230706\pi\)
0.748642 + 0.662975i \(0.230706\pi\)
\(308\) 0 0
\(309\) 6420.00 1.18195
\(310\) 0 0
\(311\) −5492.00 −1.00136 −0.500680 − 0.865633i \(-0.666917\pi\)
−0.500680 + 0.865633i \(0.666917\pi\)
\(312\) 0 0
\(313\) 422.000 0.0762072 0.0381036 − 0.999274i \(-0.487868\pi\)
0.0381036 + 0.999274i \(0.487868\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6194.00 −1.09744 −0.548722 − 0.836005i \(-0.684885\pi\)
−0.548722 + 0.836005i \(0.684885\pi\)
\(318\) 0 0
\(319\) 2272.00 0.398770
\(320\) 0 0
\(321\) −8500.00 −1.47796
\(322\) 0 0
\(323\) 744.000 0.128165
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 6660.00 1.12630
\(328\) 0 0
\(329\) −684.000 −0.114620
\(330\) 0 0
\(331\) −7688.00 −1.27665 −0.638324 − 0.769768i \(-0.720372\pi\)
−0.638324 + 0.769768i \(0.720372\pi\)
\(332\) 0 0
\(333\) −14746.0 −2.42665
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1438.00 0.232442 0.116221 − 0.993223i \(-0.462922\pi\)
0.116221 + 0.993223i \(0.462922\pi\)
\(338\) 0 0
\(339\) 14460.0 2.31669
\(340\) 0 0
\(341\) 3008.00 0.477690
\(342\) 0 0
\(343\) 6516.00 1.02575
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 8838.00 1.36729 0.683644 − 0.729816i \(-0.260394\pi\)
0.683644 + 0.729816i \(0.260394\pi\)
\(348\) 0 0
\(349\) −7810.00 −1.19788 −0.598939 − 0.800794i \(-0.704411\pi\)
−0.598939 + 0.800794i \(0.704411\pi\)
\(350\) 0 0
\(351\) 2760.00 0.419709
\(352\) 0 0
\(353\) −5906.00 −0.890495 −0.445247 − 0.895408i \(-0.646884\pi\)
−0.445247 + 0.895408i \(0.646884\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −1080.00 −0.160111
\(358\) 0 0
\(359\) −8904.00 −1.30901 −0.654506 − 0.756057i \(-0.727123\pi\)
−0.654506 + 0.756057i \(0.727123\pi\)
\(360\) 0 0
\(361\) 8517.00 1.24173
\(362\) 0 0
\(363\) −10750.0 −1.55435
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −7370.00 −1.04826 −0.524129 − 0.851639i \(-0.675609\pi\)
−0.524129 + 0.851639i \(0.675609\pi\)
\(368\) 0 0
\(369\) 3942.00 0.556131
\(370\) 0 0
\(371\) 13284.0 1.85895
\(372\) 0 0
\(373\) 734.000 0.101890 0.0509451 − 0.998701i \(-0.483777\pi\)
0.0509451 + 0.998701i \(0.483777\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 852.000 0.116393
\(378\) 0 0
\(379\) −10300.0 −1.39598 −0.697989 − 0.716109i \(-0.745921\pi\)
−0.697989 + 0.716109i \(0.745921\pi\)
\(380\) 0 0
\(381\) −11540.0 −1.55174
\(382\) 0 0
\(383\) 2682.00 0.357817 0.178908 − 0.983866i \(-0.442743\pi\)
0.178908 + 0.983866i \(0.442743\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4818.00 0.632849
\(388\) 0 0
\(389\) 6114.00 0.796895 0.398447 − 0.917191i \(-0.369549\pi\)
0.398447 + 0.917191i \(0.369549\pi\)
\(390\) 0 0
\(391\) 252.000 0.0325938
\(392\) 0 0
\(393\) 3680.00 0.472345
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 7174.00 0.906934 0.453467 − 0.891273i \(-0.350187\pi\)
0.453467 + 0.891273i \(0.350187\pi\)
\(398\) 0 0
\(399\) −22320.0 −2.80050
\(400\) 0 0
\(401\) 10498.0 1.30734 0.653672 − 0.756778i \(-0.273228\pi\)
0.653672 + 0.756778i \(0.273228\pi\)
\(402\) 0 0
\(403\) 1128.00 0.139428
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −3232.00 −0.393622
\(408\) 0 0
\(409\) −1810.00 −0.218823 −0.109412 − 0.993997i \(-0.534897\pi\)
−0.109412 + 0.993997i \(0.534897\pi\)
\(410\) 0 0
\(411\) 6700.00 0.804104
\(412\) 0 0
\(413\) 10152.0 1.20956
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 5720.00 0.671726
\(418\) 0 0
\(419\) 3396.00 0.395956 0.197978 − 0.980206i \(-0.436563\pi\)
0.197978 + 0.980206i \(0.436563\pi\)
\(420\) 0 0
\(421\) −14974.0 −1.73346 −0.866732 − 0.498775i \(-0.833784\pi\)
−0.866732 + 0.498775i \(0.833784\pi\)
\(422\) 0 0
\(423\) 2774.00 0.318857
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 4716.00 0.534481
\(428\) 0 0
\(429\) 960.000 0.108040
\(430\) 0 0
\(431\) 13540.0 1.51322 0.756611 − 0.653865i \(-0.226854\pi\)
0.756611 + 0.653865i \(0.226854\pi\)
\(432\) 0 0
\(433\) −15426.0 −1.71207 −0.856035 − 0.516918i \(-0.827079\pi\)
−0.856035 + 0.516918i \(0.827079\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 5208.00 0.570097
\(438\) 0 0
\(439\) 10472.0 1.13850 0.569250 − 0.822165i \(-0.307234\pi\)
0.569250 + 0.822165i \(0.307234\pi\)
\(440\) 0 0
\(441\) −1387.00 −0.149768
\(442\) 0 0
\(443\) 722.000 0.0774340 0.0387170 − 0.999250i \(-0.487673\pi\)
0.0387170 + 0.999250i \(0.487673\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 17300.0 1.83056
\(448\) 0 0
\(449\) −11898.0 −1.25056 −0.625280 − 0.780401i \(-0.715015\pi\)
−0.625280 + 0.780401i \(0.715015\pi\)
\(450\) 0 0
\(451\) 864.000 0.0902088
\(452\) 0 0
\(453\) −13240.0 −1.37322
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 790.000 0.0808635 0.0404318 − 0.999182i \(-0.487127\pi\)
0.0404318 + 0.999182i \(0.487127\pi\)
\(458\) 0 0
\(459\) 2760.00 0.280666
\(460\) 0 0
\(461\) −3418.00 −0.345319 −0.172660 − 0.984982i \(-0.555236\pi\)
−0.172660 + 0.984982i \(0.555236\pi\)
\(462\) 0 0
\(463\) −7534.00 −0.756230 −0.378115 − 0.925759i \(-0.623428\pi\)
−0.378115 + 0.925759i \(0.623428\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −14314.0 −1.41836 −0.709179 − 0.705029i \(-0.750934\pi\)
−0.709179 + 0.705029i \(0.750934\pi\)
\(468\) 0 0
\(469\) 9972.00 0.981800
\(470\) 0 0
\(471\) −29460.0 −2.88205
\(472\) 0 0
\(473\) 1056.00 0.102653
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −53874.0 −5.17132
\(478\) 0 0
\(479\) 7016.00 0.669247 0.334623 − 0.942352i \(-0.391391\pi\)
0.334623 + 0.942352i \(0.391391\pi\)
\(480\) 0 0
\(481\) −1212.00 −0.114891
\(482\) 0 0
\(483\) −7560.00 −0.712199
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 15190.0 1.41340 0.706699 − 0.707515i \(-0.250184\pi\)
0.706699 + 0.707515i \(0.250184\pi\)
\(488\) 0 0
\(489\) 20980.0 1.94018
\(490\) 0 0
\(491\) 12624.0 1.16031 0.580156 − 0.814505i \(-0.302992\pi\)
0.580156 + 0.814505i \(0.302992\pi\)
\(492\) 0 0
\(493\) 852.000 0.0778340
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2520.00 0.227440
\(498\) 0 0
\(499\) 2492.00 0.223562 0.111781 − 0.993733i \(-0.464345\pi\)
0.111781 + 0.993733i \(0.464345\pi\)
\(500\) 0 0
\(501\) −8660.00 −0.772256
\(502\) 0 0
\(503\) 11714.0 1.03837 0.519186 − 0.854661i \(-0.326235\pi\)
0.519186 + 0.854661i \(0.326235\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −21610.0 −1.89297
\(508\) 0 0
\(509\) −5618.00 −0.489221 −0.244610 − 0.969621i \(-0.578660\pi\)
−0.244610 + 0.969621i \(0.578660\pi\)
\(510\) 0 0
\(511\) 15876.0 1.37439
\(512\) 0 0
\(513\) 57040.0 4.90912
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 608.000 0.0517211
\(518\) 0 0
\(519\) 16780.0 1.41919
\(520\) 0 0
\(521\) 13770.0 1.15792 0.578958 − 0.815357i \(-0.303459\pi\)
0.578958 + 0.815357i \(0.303459\pi\)
\(522\) 0 0
\(523\) 6986.00 0.584085 0.292042 − 0.956405i \(-0.405665\pi\)
0.292042 + 0.956405i \(0.405665\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1128.00 0.0932380
\(528\) 0 0
\(529\) −10403.0 −0.855018
\(530\) 0 0
\(531\) −41172.0 −3.36481
\(532\) 0 0
\(533\) 324.000 0.0263302
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 16200.0 1.30183
\(538\) 0 0
\(539\) −304.000 −0.0242935
\(540\) 0 0
\(541\) 11958.0 0.950304 0.475152 − 0.879904i \(-0.342393\pi\)
0.475152 + 0.879904i \(0.342393\pi\)
\(542\) 0 0
\(543\) 25100.0 1.98369
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −4194.00 −0.327829 −0.163915 − 0.986475i \(-0.552412\pi\)
−0.163915 + 0.986475i \(0.552412\pi\)
\(548\) 0 0
\(549\) −19126.0 −1.48684
\(550\) 0 0
\(551\) 17608.0 1.36139
\(552\) 0 0
\(553\) −20880.0 −1.60562
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 5382.00 0.409412 0.204706 − 0.978823i \(-0.434376\pi\)
0.204706 + 0.978823i \(0.434376\pi\)
\(558\) 0 0
\(559\) 396.000 0.0299625
\(560\) 0 0
\(561\) 960.000 0.0722482
\(562\) 0 0
\(563\) 15418.0 1.15416 0.577079 − 0.816688i \(-0.304192\pi\)
0.577079 + 0.816688i \(0.304192\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −47322.0 −3.50500
\(568\) 0 0
\(569\) −5778.00 −0.425705 −0.212853 − 0.977084i \(-0.568275\pi\)
−0.212853 + 0.977084i \(0.568275\pi\)
\(570\) 0 0
\(571\) −6024.00 −0.441500 −0.220750 − 0.975330i \(-0.570851\pi\)
−0.220750 + 0.975330i \(0.570851\pi\)
\(572\) 0 0
\(573\) 3720.00 0.271213
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −554.000 −0.0399711 −0.0199855 − 0.999800i \(-0.506362\pi\)
−0.0199855 + 0.999800i \(0.506362\pi\)
\(578\) 0 0
\(579\) −29380.0 −2.10879
\(580\) 0 0
\(581\) −11556.0 −0.825170
\(582\) 0 0
\(583\) −11808.0 −0.838829
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −2386.00 −0.167770 −0.0838848 − 0.996475i \(-0.526733\pi\)
−0.0838848 + 0.996475i \(0.526733\pi\)
\(588\) 0 0
\(589\) 23312.0 1.63082
\(590\) 0 0
\(591\) −22340.0 −1.55490
\(592\) 0 0
\(593\) 846.000 0.0585853 0.0292926 − 0.999571i \(-0.490675\pi\)
0.0292926 + 0.999571i \(0.490675\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 30480.0 2.08955
\(598\) 0 0
\(599\) −22304.0 −1.52140 −0.760698 − 0.649105i \(-0.775143\pi\)
−0.760698 + 0.649105i \(0.775143\pi\)
\(600\) 0 0
\(601\) 5510.00 0.373973 0.186986 − 0.982363i \(-0.440128\pi\)
0.186986 + 0.982363i \(0.440128\pi\)
\(602\) 0 0
\(603\) −40442.0 −2.73122
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −8234.00 −0.550589 −0.275295 − 0.961360i \(-0.588775\pi\)
−0.275295 + 0.961360i \(0.588775\pi\)
\(608\) 0 0
\(609\) −25560.0 −1.70073
\(610\) 0 0
\(611\) 228.000 0.0150964
\(612\) 0 0
\(613\) 1046.00 0.0689193 0.0344597 − 0.999406i \(-0.489029\pi\)
0.0344597 + 0.999406i \(0.489029\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3862.00 0.251991 0.125995 − 0.992031i \(-0.459788\pi\)
0.125995 + 0.992031i \(0.459788\pi\)
\(618\) 0 0
\(619\) −13964.0 −0.906721 −0.453361 − 0.891327i \(-0.649775\pi\)
−0.453361 + 0.891327i \(0.649775\pi\)
\(620\) 0 0
\(621\) 19320.0 1.24845
\(622\) 0 0
\(623\) 15372.0 0.988549
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 19840.0 1.26369
\(628\) 0 0
\(629\) −1212.00 −0.0768293
\(630\) 0 0
\(631\) 14884.0 0.939022 0.469511 − 0.882927i \(-0.344430\pi\)
0.469511 + 0.882927i \(0.344430\pi\)
\(632\) 0 0
\(633\) −48960.0 −3.07423
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −114.000 −0.00709081
\(638\) 0 0
\(639\) −10220.0 −0.632703
\(640\) 0 0
\(641\) 17838.0 1.09916 0.549578 − 0.835443i \(-0.314789\pi\)
0.549578 + 0.835443i \(0.314789\pi\)
\(642\) 0 0
\(643\) −7814.00 −0.479244 −0.239622 − 0.970866i \(-0.577024\pi\)
−0.239622 + 0.970866i \(0.577024\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 774.000 0.0470310 0.0235155 − 0.999723i \(-0.492514\pi\)
0.0235155 + 0.999723i \(0.492514\pi\)
\(648\) 0 0
\(649\) −9024.00 −0.545798
\(650\) 0 0
\(651\) −33840.0 −2.03732
\(652\) 0 0
\(653\) 23422.0 1.40364 0.701818 − 0.712357i \(-0.252372\pi\)
0.701818 + 0.712357i \(0.252372\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −64386.0 −3.82334
\(658\) 0 0
\(659\) 13508.0 0.798478 0.399239 − 0.916847i \(-0.369274\pi\)
0.399239 + 0.916847i \(0.369274\pi\)
\(660\) 0 0
\(661\) −6222.00 −0.366124 −0.183062 − 0.983101i \(-0.558601\pi\)
−0.183062 + 0.983101i \(0.558601\pi\)
\(662\) 0 0
\(663\) 360.000 0.0210878
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 5964.00 0.346217
\(668\) 0 0
\(669\) −53020.0 −3.06408
\(670\) 0 0
\(671\) −4192.00 −0.241178
\(672\) 0 0
\(673\) 15566.0 0.891568 0.445784 − 0.895141i \(-0.352925\pi\)
0.445784 + 0.895141i \(0.352925\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −2234.00 −0.126824 −0.0634118 − 0.997987i \(-0.520198\pi\)
−0.0634118 + 0.997987i \(0.520198\pi\)
\(678\) 0 0
\(679\) −8604.00 −0.486290
\(680\) 0 0
\(681\) −37780.0 −2.12589
\(682\) 0 0
\(683\) 13282.0 0.744102 0.372051 − 0.928212i \(-0.378655\pi\)
0.372051 + 0.928212i \(0.378655\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −30340.0 −1.68492
\(688\) 0 0
\(689\) −4428.00 −0.244838
\(690\) 0 0
\(691\) 27416.0 1.50934 0.754670 − 0.656105i \(-0.227797\pi\)
0.754670 + 0.656105i \(0.227797\pi\)
\(692\) 0 0
\(693\) −21024.0 −1.15243
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 324.000 0.0176074
\(698\) 0 0
\(699\) 34780.0 1.88197
\(700\) 0 0
\(701\) 25626.0 1.38071 0.690357 − 0.723469i \(-0.257453\pi\)
0.690357 + 0.723469i \(0.257453\pi\)
\(702\) 0 0
\(703\) −25048.0 −1.34382
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 32292.0 1.71777
\(708\) 0 0
\(709\) 11702.0 0.619856 0.309928 − 0.950760i \(-0.399695\pi\)
0.309928 + 0.950760i \(0.399695\pi\)
\(710\) 0 0
\(711\) 84680.0 4.46659
\(712\) 0 0
\(713\) 7896.00 0.414737
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −15600.0 −0.812542
\(718\) 0 0
\(719\) 28008.0 1.45274 0.726371 − 0.687302i \(-0.241205\pi\)
0.726371 + 0.687302i \(0.241205\pi\)
\(720\) 0 0
\(721\) −11556.0 −0.596904
\(722\) 0 0
\(723\) −32180.0 −1.65531
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −7682.00 −0.391898 −0.195949 − 0.980614i \(-0.562779\pi\)
−0.195949 + 0.980614i \(0.562779\pi\)
\(728\) 0 0
\(729\) 67717.0 3.44038
\(730\) 0 0
\(731\) 396.000 0.0200364
\(732\) 0 0
\(733\) 14270.0 0.719065 0.359532 − 0.933133i \(-0.382936\pi\)
0.359532 + 0.933133i \(0.382936\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −8864.00 −0.443025
\(738\) 0 0
\(739\) −29324.0 −1.45968 −0.729838 − 0.683620i \(-0.760405\pi\)
−0.729838 + 0.683620i \(0.760405\pi\)
\(740\) 0 0
\(741\) 7440.00 0.368846
\(742\) 0 0
\(743\) 29258.0 1.44465 0.722323 − 0.691556i \(-0.243074\pi\)
0.722323 + 0.691556i \(0.243074\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 46866.0 2.29550
\(748\) 0 0
\(749\) 15300.0 0.746395
\(750\) 0 0
\(751\) −19076.0 −0.926888 −0.463444 − 0.886126i \(-0.653387\pi\)
−0.463444 + 0.886126i \(0.653387\pi\)
\(752\) 0 0
\(753\) −6880.00 −0.332963
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 22670.0 1.08845 0.544224 − 0.838940i \(-0.316824\pi\)
0.544224 + 0.838940i \(0.316824\pi\)
\(758\) 0 0
\(759\) 6720.00 0.321371
\(760\) 0 0
\(761\) −23206.0 −1.10541 −0.552705 − 0.833377i \(-0.686404\pi\)
−0.552705 + 0.833377i \(0.686404\pi\)
\(762\) 0 0
\(763\) −11988.0 −0.568800
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −3384.00 −0.159308
\(768\) 0 0
\(769\) −1854.00 −0.0869401 −0.0434701 − 0.999055i \(-0.513841\pi\)
−0.0434701 + 0.999055i \(0.513841\pi\)
\(770\) 0 0
\(771\) −21700.0 −1.01363
\(772\) 0 0
\(773\) −6474.00 −0.301234 −0.150617 − 0.988592i \(-0.548126\pi\)
−0.150617 + 0.988592i \(0.548126\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 36360.0 1.67877
\(778\) 0 0
\(779\) 6696.00 0.307971
\(780\) 0 0
\(781\) −2240.00 −0.102629
\(782\) 0 0
\(783\) 65320.0 2.98129
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −20354.0 −0.921908 −0.460954 − 0.887424i \(-0.652493\pi\)
−0.460954 + 0.887424i \(0.652493\pi\)
\(788\) 0 0
\(789\) 22740.0 1.02607
\(790\) 0 0
\(791\) −26028.0 −1.16997
\(792\) 0 0
\(793\) −1572.00 −0.0703952
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1886.00 0.0838213 0.0419106 − 0.999121i \(-0.486656\pi\)
0.0419106 + 0.999121i \(0.486656\pi\)
\(798\) 0 0
\(799\) 228.000 0.0100952
\(800\) 0 0
\(801\) −62342.0 −2.75000
\(802\) 0 0
\(803\) −14112.0 −0.620176
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 71460.0 3.11711
\(808\) 0 0
\(809\) −9462.00 −0.411207 −0.205603 − 0.978635i \(-0.565916\pi\)
−0.205603 + 0.978635i \(0.565916\pi\)
\(810\) 0 0
\(811\) −24512.0 −1.06132 −0.530661 − 0.847584i \(-0.678056\pi\)
−0.530661 + 0.847584i \(0.678056\pi\)
\(812\) 0 0
\(813\) −26040.0 −1.12332
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 8184.00 0.350455
\(818\) 0 0
\(819\) −7884.00 −0.336373
\(820\) 0 0
\(821\) 36242.0 1.54063 0.770313 − 0.637666i \(-0.220100\pi\)
0.770313 + 0.637666i \(0.220100\pi\)
\(822\) 0 0
\(823\) −17718.0 −0.750438 −0.375219 − 0.926936i \(-0.622433\pi\)
−0.375219 + 0.926936i \(0.622433\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 6726.00 0.282812 0.141406 − 0.989952i \(-0.454838\pi\)
0.141406 + 0.989952i \(0.454838\pi\)
\(828\) 0 0
\(829\) 41722.0 1.74797 0.873984 − 0.485955i \(-0.161528\pi\)
0.873984 + 0.485955i \(0.161528\pi\)
\(830\) 0 0
\(831\) 51500.0 2.14984
\(832\) 0 0
\(833\) −114.000 −0.00474174
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 86480.0 3.57131
\(838\) 0 0
\(839\) −16720.0 −0.688008 −0.344004 − 0.938968i \(-0.611783\pi\)
−0.344004 + 0.938968i \(0.611783\pi\)
\(840\) 0 0
\(841\) −4225.00 −0.173234
\(842\) 0 0
\(843\) 52700.0 2.15313
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 19350.0 0.784975
\(848\) 0 0
\(849\) 34340.0 1.38816
\(850\) 0 0
\(851\) −8484.00 −0.341748
\(852\) 0 0
\(853\) 33286.0 1.33610 0.668049 − 0.744118i \(-0.267130\pi\)
0.668049 + 0.744118i \(0.267130\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −38978.0 −1.55363 −0.776816 − 0.629727i \(-0.783167\pi\)
−0.776816 + 0.629727i \(0.783167\pi\)
\(858\) 0 0
\(859\) 1916.00 0.0761037 0.0380518 − 0.999276i \(-0.487885\pi\)
0.0380518 + 0.999276i \(0.487885\pi\)
\(860\) 0 0
\(861\) −9720.00 −0.384735
\(862\) 0 0
\(863\) −2374.00 −0.0936407 −0.0468203 − 0.998903i \(-0.514909\pi\)
−0.0468203 + 0.998903i \(0.514909\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −48770.0 −1.91040
\(868\) 0 0
\(869\) 18560.0 0.724517
\(870\) 0 0
\(871\) −3324.00 −0.129310
\(872\) 0 0
\(873\) 34894.0 1.35279
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −32722.0 −1.25991 −0.629956 − 0.776631i \(-0.716927\pi\)
−0.629956 + 0.776631i \(0.716927\pi\)
\(878\) 0 0
\(879\) 98780.0 3.79041
\(880\) 0 0
\(881\) 5390.00 0.206122 0.103061 − 0.994675i \(-0.467136\pi\)
0.103061 + 0.994675i \(0.467136\pi\)
\(882\) 0 0
\(883\) −43238.0 −1.64788 −0.823938 − 0.566680i \(-0.808228\pi\)
−0.823938 + 0.566680i \(0.808228\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −11010.0 −0.416775 −0.208388 − 0.978046i \(-0.566822\pi\)
−0.208388 + 0.978046i \(0.566822\pi\)
\(888\) 0 0
\(889\) 20772.0 0.783656
\(890\) 0 0
\(891\) 42064.0 1.58159
\(892\) 0 0
\(893\) 4712.00 0.176575
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 2520.00 0.0938020
\(898\) 0 0
\(899\) 26696.0 0.990391
\(900\) 0 0
\(901\) −4428.00 −0.163727
\(902\) 0 0
\(903\) −11880.0 −0.437809
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −74.0000 −0.00270907 −0.00135454 − 0.999999i \(-0.500431\pi\)
−0.00135454 + 0.999999i \(0.500431\pi\)
\(908\) 0 0
\(909\) −130962. −4.77859
\(910\) 0 0
\(911\) −17460.0 −0.634990 −0.317495 − 0.948260i \(-0.602842\pi\)
−0.317495 + 0.948260i \(0.602842\pi\)
\(912\) 0 0
\(913\) 10272.0 0.372348
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −6624.00 −0.238543
\(918\) 0 0
\(919\) −17072.0 −0.612789 −0.306395 − 0.951905i \(-0.599123\pi\)
−0.306395 + 0.951905i \(0.599123\pi\)
\(920\) 0 0
\(921\) 80540.0 2.88152
\(922\) 0 0
\(923\) −840.000 −0.0299555
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 46866.0 1.66050
\(928\) 0 0
\(929\) −14826.0 −0.523601 −0.261800 − 0.965122i \(-0.584316\pi\)
−0.261800 + 0.965122i \(0.584316\pi\)
\(930\) 0 0
\(931\) −2356.00 −0.0829375
\(932\) 0 0
\(933\) −54920.0 −1.92712
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −3354.00 −0.116937 −0.0584687 − 0.998289i \(-0.518622\pi\)
−0.0584687 + 0.998289i \(0.518622\pi\)
\(938\) 0 0
\(939\) 4220.00 0.146661
\(940\) 0 0
\(941\) −15434.0 −0.534680 −0.267340 − 0.963602i \(-0.586145\pi\)
−0.267340 + 0.963602i \(0.586145\pi\)
\(942\) 0 0
\(943\) 2268.00 0.0783205
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −9306.00 −0.319329 −0.159664 − 0.987171i \(-0.551041\pi\)
−0.159664 + 0.987171i \(0.551041\pi\)
\(948\) 0 0
\(949\) −5292.00 −0.181017
\(950\) 0 0
\(951\) −61940.0 −2.11203
\(952\) 0 0
\(953\) −12202.0 −0.414755 −0.207378 − 0.978261i \(-0.566493\pi\)
−0.207378 + 0.978261i \(0.566493\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 22720.0 0.767433
\(958\) 0 0
\(959\) −12060.0 −0.406087
\(960\) 0 0
\(961\) 5553.00 0.186399
\(962\) 0 0
\(963\) −62050.0 −2.07636
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 17478.0 0.581235 0.290618 − 0.956839i \(-0.406139\pi\)
0.290618 + 0.956839i \(0.406139\pi\)
\(968\) 0 0
\(969\) 7440.00 0.246653
\(970\) 0 0
\(971\) −10920.0 −0.360906 −0.180453 − 0.983584i \(-0.557756\pi\)
−0.180453 + 0.983584i \(0.557756\pi\)
\(972\) 0 0
\(973\) −10296.0 −0.339234
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −10834.0 −0.354770 −0.177385 − 0.984142i \(-0.556764\pi\)
−0.177385 + 0.984142i \(0.556764\pi\)
\(978\) 0 0
\(979\) −13664.0 −0.446071
\(980\) 0 0
\(981\) 48618.0 1.58232
\(982\) 0 0
\(983\) −36862.0 −1.19605 −0.598024 − 0.801478i \(-0.704047\pi\)
−0.598024 + 0.801478i \(0.704047\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −6840.00 −0.220587
\(988\) 0 0
\(989\) 2772.00 0.0891248
\(990\) 0 0
\(991\) −5380.00 −0.172453 −0.0862267 − 0.996276i \(-0.527481\pi\)
−0.0862267 + 0.996276i \(0.527481\pi\)
\(992\) 0 0
\(993\) −76880.0 −2.45691
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −31266.0 −0.993184 −0.496592 − 0.867984i \(-0.665415\pi\)
−0.496592 + 0.867984i \(0.665415\pi\)
\(998\) 0 0
\(999\) −92920.0 −2.94280
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 400.4.a.u.1.1 1
4.3 odd 2 200.4.a.a.1.1 1
5.2 odd 4 400.4.c.a.49.1 2
5.3 odd 4 400.4.c.a.49.2 2
5.4 even 2 80.4.a.a.1.1 1
8.3 odd 2 1600.4.a.ca.1.1 1
8.5 even 2 1600.4.a.a.1.1 1
12.11 even 2 1800.4.a.bd.1.1 1
15.14 odd 2 720.4.a.ba.1.1 1
20.3 even 4 200.4.c.a.49.1 2
20.7 even 4 200.4.c.a.49.2 2
20.19 odd 2 40.4.a.c.1.1 ✓ 1
40.19 odd 2 320.4.a.a.1.1 1
40.29 even 2 320.4.a.n.1.1 1
60.23 odd 4 1800.4.f.n.649.1 2
60.47 odd 4 1800.4.f.n.649.2 2
60.59 even 2 360.4.a.i.1.1 1
80.19 odd 4 1280.4.d.o.641.2 2
80.29 even 4 1280.4.d.b.641.1 2
80.59 odd 4 1280.4.d.o.641.1 2
80.69 even 4 1280.4.d.b.641.2 2
140.139 even 2 1960.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.a.c.1.1 ✓ 1 20.19 odd 2
80.4.a.a.1.1 1 5.4 even 2
200.4.a.a.1.1 1 4.3 odd 2
200.4.c.a.49.1 2 20.3 even 4
200.4.c.a.49.2 2 20.7 even 4
320.4.a.a.1.1 1 40.19 odd 2
320.4.a.n.1.1 1 40.29 even 2
360.4.a.i.1.1 1 60.59 even 2
400.4.a.u.1.1 1 1.1 even 1 trivial
400.4.c.a.49.1 2 5.2 odd 4
400.4.c.a.49.2 2 5.3 odd 4
720.4.a.ba.1.1 1 15.14 odd 2
1280.4.d.b.641.1 2 80.29 even 4
1280.4.d.b.641.2 2 80.69 even 4
1280.4.d.o.641.1 2 80.59 odd 4
1280.4.d.o.641.2 2 80.19 odd 4
1600.4.a.a.1.1 1 8.5 even 2
1600.4.a.ca.1.1 1 8.3 odd 2
1800.4.a.bd.1.1 1 12.11 even 2
1800.4.f.n.649.1 2 60.23 odd 4
1800.4.f.n.649.2 2 60.47 odd 4
1960.4.a.a.1.1 1 140.139 even 2