Properties

Label 900.6.a.u.1.2
Level $900$
Weight $6$
Character 900.1
Self dual yes
Analytic conductor $144.345$
Analytic rank $0$
Dimension $2$
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] = [900,6,Mod(1,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 900.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: \(144.345437832\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{241}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 60 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 180)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-7.26209\) of defining polynomial
Character \(\chi\) \(=\) 900.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+133.145 q^{7} +O(q^{10})\) \(q+133.145 q^{7} -345.725 q^{11} -405.160 q^{13} -2132.90 q^{17} -1314.90 q^{19} -808.550 q^{23} +984.198 q^{29} +8016.15 q^{31} +9348.70 q^{37} +19148.7 q^{41} +5836.76 q^{43} -8894.04 q^{47} +920.604 q^{49} -30515.8 q^{53} +49014.9 q^{59} -423.406 q^{61} +10469.8 q^{67} -7817.87 q^{71} -27144.8 q^{73} -46031.6 q^{77} +31564.2 q^{79} -8754.66 q^{83} -21540.9 q^{89} -53945.1 q^{91} -52697.0 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 80 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 80 q^{7} + 240 q^{11} + 680 q^{13} - 540 q^{17} + 1096 q^{19} - 3480 q^{23} + 9420 q^{29} + 2992 q^{31} + 7520 q^{37} + 27120 q^{41} - 4720 q^{43} - 38280 q^{47} - 13062 q^{49} - 53580 q^{53} + 46800 q^{59} + 21508 q^{61} + 10880 q^{67} + 45840 q^{71} - 9580 q^{73} - 77160 q^{77} + 91072 q^{79} + 14160 q^{83} + 35160 q^{89} - 111616 q^{91} + 98780 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 133.145 1.02702 0.513511 0.858083i \(-0.328344\pi\)
0.513511 + 0.858083i \(0.328344\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −345.725 −0.861488 −0.430744 0.902474i \(-0.641749\pi\)
−0.430744 + 0.902474i \(0.641749\pi\)
\(12\) 0 0
\(13\) −405.160 −0.664919 −0.332459 0.943118i \(-0.607878\pi\)
−0.332459 + 0.943118i \(0.607878\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2132.90 −1.78998 −0.894990 0.446085i \(-0.852818\pi\)
−0.894990 + 0.446085i \(0.852818\pi\)
\(18\) 0 0
\(19\) −1314.90 −0.835620 −0.417810 0.908534i \(-0.637202\pi\)
−0.417810 + 0.908534i \(0.637202\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −808.550 −0.318704 −0.159352 0.987222i \(-0.550940\pi\)
−0.159352 + 0.987222i \(0.550940\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 984.198 0.217314 0.108657 0.994079i \(-0.465345\pi\)
0.108657 + 0.994079i \(0.465345\pi\)
\(30\) 0 0
\(31\) 8016.15 1.49817 0.749086 0.662473i \(-0.230493\pi\)
0.749086 + 0.662473i \(0.230493\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 9348.70 1.12266 0.561328 0.827593i \(-0.310290\pi\)
0.561328 + 0.827593i \(0.310290\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 19148.7 1.77902 0.889508 0.456920i \(-0.151047\pi\)
0.889508 + 0.456920i \(0.151047\pi\)
\(42\) 0 0
\(43\) 5836.76 0.481394 0.240697 0.970600i \(-0.422624\pi\)
0.240697 + 0.970600i \(0.422624\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8894.04 −0.587293 −0.293646 0.955914i \(-0.594869\pi\)
−0.293646 + 0.955914i \(0.594869\pi\)
\(48\) 0 0
\(49\) 920.604 0.0547750
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −30515.8 −1.49223 −0.746114 0.665818i \(-0.768083\pi\)
−0.746114 + 0.665818i \(0.768083\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 49014.9 1.83315 0.916575 0.399863i \(-0.130942\pi\)
0.916575 + 0.399863i \(0.130942\pi\)
\(60\) 0 0
\(61\) −423.406 −0.0145691 −0.00728454 0.999973i \(-0.502319\pi\)
−0.00728454 + 0.999973i \(0.502319\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 10469.8 0.284940 0.142470 0.989799i \(-0.454496\pi\)
0.142470 + 0.989799i \(0.454496\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −7817.87 −0.184053 −0.0920264 0.995757i \(-0.529334\pi\)
−0.0920264 + 0.995757i \(0.529334\pi\)
\(72\) 0 0
\(73\) −27144.8 −0.596183 −0.298092 0.954537i \(-0.596350\pi\)
−0.298092 + 0.954537i \(0.596350\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −46031.6 −0.884768
\(78\) 0 0
\(79\) 31564.2 0.569020 0.284510 0.958673i \(-0.408169\pi\)
0.284510 + 0.958673i \(0.408169\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −8754.66 −0.139490 −0.0697451 0.997565i \(-0.522219\pi\)
−0.0697451 + 0.997565i \(0.522219\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −21540.9 −0.288263 −0.144132 0.989559i \(-0.546039\pi\)
−0.144132 + 0.989559i \(0.546039\pi\)
\(90\) 0 0
\(91\) −53945.1 −0.682886
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −52697.0 −0.568665 −0.284332 0.958726i \(-0.591772\pi\)
−0.284332 + 0.958726i \(0.591772\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 72895.0 0.711041 0.355520 0.934668i \(-0.384304\pi\)
0.355520 + 0.934668i \(0.384304\pi\)
\(102\) 0 0
\(103\) 2860.23 0.0265649 0.0132824 0.999912i \(-0.495772\pi\)
0.0132824 + 0.999912i \(0.495772\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 120948. 1.02127 0.510633 0.859799i \(-0.329411\pi\)
0.510633 + 0.859799i \(0.329411\pi\)
\(108\) 0 0
\(109\) 41121.4 0.331514 0.165757 0.986167i \(-0.446993\pi\)
0.165757 + 0.986167i \(0.446993\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 138639. 1.02138 0.510691 0.859765i \(-0.329390\pi\)
0.510691 + 0.859765i \(0.329390\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −283985. −1.83835
\(120\) 0 0
\(121\) −41525.1 −0.257838
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 253834. 1.39650 0.698250 0.715854i \(-0.253962\pi\)
0.698250 + 0.715854i \(0.253962\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −360926. −1.83755 −0.918776 0.394779i \(-0.870821\pi\)
−0.918776 + 0.394779i \(0.870821\pi\)
\(132\) 0 0
\(133\) −175073. −0.858201
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2885.73 −0.0131357 −0.00656787 0.999978i \(-0.502091\pi\)
−0.00656787 + 0.999978i \(0.502091\pi\)
\(138\) 0 0
\(139\) 422399. 1.85433 0.927163 0.374659i \(-0.122240\pi\)
0.927163 + 0.374659i \(0.122240\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 140074. 0.572820
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 318577. 1.17557 0.587786 0.809016i \(-0.300000\pi\)
0.587786 + 0.809016i \(0.300000\pi\)
\(150\) 0 0
\(151\) 186709. 0.666382 0.333191 0.942859i \(-0.391875\pi\)
0.333191 + 0.942859i \(0.391875\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 542892. 1.75778 0.878889 0.477026i \(-0.158285\pi\)
0.878889 + 0.477026i \(0.158285\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −107654. −0.327316
\(162\) 0 0
\(163\) 159620. 0.470563 0.235282 0.971927i \(-0.424399\pi\)
0.235282 + 0.971927i \(0.424399\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 411329. 1.14130 0.570648 0.821195i \(-0.306692\pi\)
0.570648 + 0.821195i \(0.306692\pi\)
\(168\) 0 0
\(169\) −207138. −0.557883
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −172542. −0.438308 −0.219154 0.975690i \(-0.570330\pi\)
−0.219154 + 0.975690i \(0.570330\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −404175. −0.942837 −0.471418 0.881910i \(-0.656258\pi\)
−0.471418 + 0.881910i \(0.656258\pi\)
\(180\) 0 0
\(181\) 713123. 1.61796 0.808981 0.587835i \(-0.200020\pi\)
0.808981 + 0.587835i \(0.200020\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 737398. 1.54205
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 523963. 1.03924 0.519622 0.854396i \(-0.326073\pi\)
0.519622 + 0.854396i \(0.326073\pi\)
\(192\) 0 0
\(193\) −575430. −1.11199 −0.555993 0.831187i \(-0.687662\pi\)
−0.555993 + 0.831187i \(0.687662\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −201175. −0.369326 −0.184663 0.982802i \(-0.559119\pi\)
−0.184663 + 0.982802i \(0.559119\pi\)
\(198\) 0 0
\(199\) 928979. 1.66293 0.831464 0.555579i \(-0.187503\pi\)
0.831464 + 0.555579i \(0.187503\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 131041. 0.223186
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 454594. 0.719877
\(210\) 0 0
\(211\) −541317. −0.837038 −0.418519 0.908208i \(-0.637451\pi\)
−0.418519 + 0.908208i \(0.637451\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.06731e6 1.53866
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 864167. 1.19019
\(222\) 0 0
\(223\) −852442. −1.14790 −0.573948 0.818892i \(-0.694589\pi\)
−0.573948 + 0.818892i \(0.694589\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.00310e6 −1.29206 −0.646029 0.763313i \(-0.723571\pi\)
−0.646029 + 0.763313i \(0.723571\pi\)
\(228\) 0 0
\(229\) 1.21171e6 1.52690 0.763448 0.645869i \(-0.223505\pi\)
0.763448 + 0.645869i \(0.223505\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.26704e6 −1.52898 −0.764488 0.644638i \(-0.777008\pi\)
−0.764488 + 0.644638i \(0.777008\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 570056. 0.645539 0.322770 0.946478i \(-0.395386\pi\)
0.322770 + 0.946478i \(0.395386\pi\)
\(240\) 0 0
\(241\) −396643. −0.439904 −0.219952 0.975511i \(-0.570590\pi\)
−0.219952 + 0.975511i \(0.570590\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 532746. 0.555620
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.35590e6 −1.35845 −0.679225 0.733930i \(-0.737684\pi\)
−0.679225 + 0.733930i \(0.737684\pi\)
\(252\) 0 0
\(253\) 279536. 0.274559
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.86712e6 −1.76336 −0.881679 0.471850i \(-0.843586\pi\)
−0.881679 + 0.471850i \(0.843586\pi\)
\(258\) 0 0
\(259\) 1.24473e6 1.15299
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.03269e6 0.920618 0.460309 0.887759i \(-0.347739\pi\)
0.460309 + 0.887759i \(0.347739\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 189382. 0.159573 0.0797863 0.996812i \(-0.474576\pi\)
0.0797863 + 0.996812i \(0.474576\pi\)
\(270\) 0 0
\(271\) −584853. −0.483752 −0.241876 0.970307i \(-0.577763\pi\)
−0.241876 + 0.970307i \(0.577763\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.45412e6 1.13867 0.569337 0.822104i \(-0.307200\pi\)
0.569337 + 0.822104i \(0.307200\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.05546e6 0.797399 0.398700 0.917082i \(-0.369462\pi\)
0.398700 + 0.917082i \(0.369462\pi\)
\(282\) 0 0
\(283\) −494652. −0.367142 −0.183571 0.983006i \(-0.558766\pi\)
−0.183571 + 0.983006i \(0.558766\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.54955e6 1.82709
\(288\) 0 0
\(289\) 3.12941e6 2.20403
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2.23259e6 −1.51929 −0.759643 0.650340i \(-0.774627\pi\)
−0.759643 + 0.650340i \(0.774627\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 327592. 0.211912
\(300\) 0 0
\(301\) 777136. 0.494403
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 2.91085e6 1.76268 0.881342 0.472478i \(-0.156641\pi\)
0.881342 + 0.472478i \(0.156641\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1.47978e6 0.867554 0.433777 0.901020i \(-0.357181\pi\)
0.433777 + 0.901020i \(0.357181\pi\)
\(312\) 0 0
\(313\) −1.83025e6 −1.05596 −0.527981 0.849256i \(-0.677051\pi\)
−0.527981 + 0.849256i \(0.677051\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.22760e6 1.24506 0.622528 0.782598i \(-0.286106\pi\)
0.622528 + 0.782598i \(0.286106\pi\)
\(318\) 0 0
\(319\) −340262. −0.187213
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.80455e6 1.49574
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −1.18420e6 −0.603163
\(330\) 0 0
\(331\) 2.63588e6 1.32238 0.661188 0.750220i \(-0.270052\pi\)
0.661188 + 0.750220i \(0.270052\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.28650e6 1.09672 0.548361 0.836242i \(-0.315252\pi\)
0.548361 + 0.836242i \(0.315252\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −2.77139e6 −1.29066
\(342\) 0 0
\(343\) −2.11519e6 −0.970767
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −154685. −0.0689642 −0.0344821 0.999405i \(-0.510978\pi\)
−0.0344821 + 0.999405i \(0.510978\pi\)
\(348\) 0 0
\(349\) −374471. −0.164571 −0.0822857 0.996609i \(-0.526222\pi\)
−0.0822857 + 0.996609i \(0.526222\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 3.15707e6 1.34849 0.674244 0.738508i \(-0.264469\pi\)
0.674244 + 0.738508i \(0.264469\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −3.31411e6 −1.35716 −0.678579 0.734527i \(-0.737404\pi\)
−0.678579 + 0.734527i \(0.737404\pi\)
\(360\) 0 0
\(361\) −747134. −0.301739
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 4.35079e6 1.68618 0.843088 0.537775i \(-0.180735\pi\)
0.843088 + 0.537775i \(0.180735\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −4.06303e6 −1.53255
\(372\) 0 0
\(373\) 2.72909e6 1.01565 0.507826 0.861460i \(-0.330449\pi\)
0.507826 + 0.861460i \(0.330449\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −398758. −0.144496
\(378\) 0 0
\(379\) 2.60140e6 0.930271 0.465135 0.885240i \(-0.346006\pi\)
0.465135 + 0.885240i \(0.346006\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −4.11137e6 −1.43215 −0.716076 0.698022i \(-0.754064\pi\)
−0.716076 + 0.698022i \(0.754064\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.16499e6 0.390345 0.195173 0.980769i \(-0.437473\pi\)
0.195173 + 0.980769i \(0.437473\pi\)
\(390\) 0 0
\(391\) 1.72456e6 0.570473
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 1.09268e6 0.347949 0.173975 0.984750i \(-0.444339\pi\)
0.173975 + 0.984750i \(0.444339\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 115723. 0.0359383 0.0179691 0.999839i \(-0.494280\pi\)
0.0179691 + 0.999839i \(0.494280\pi\)
\(402\) 0 0
\(403\) −3.24783e6 −0.996163
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −3.23208e6 −0.967156
\(408\) 0 0
\(409\) 4.82519e6 1.42628 0.713142 0.701020i \(-0.247271\pi\)
0.713142 + 0.701020i \(0.247271\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 6.52609e6 1.88269
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −807583. −0.224725 −0.112363 0.993667i \(-0.535842\pi\)
−0.112363 + 0.993667i \(0.535842\pi\)
\(420\) 0 0
\(421\) 2.21235e6 0.608342 0.304171 0.952617i \(-0.401621\pi\)
0.304171 + 0.952617i \(0.401621\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −56374.4 −0.0149628
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 2.29654e6 0.595499 0.297749 0.954644i \(-0.403764\pi\)
0.297749 + 0.954644i \(0.403764\pi\)
\(432\) 0 0
\(433\) 5.66261e6 1.45143 0.725716 0.687995i \(-0.241509\pi\)
0.725716 + 0.687995i \(0.241509\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.06316e6 0.266315
\(438\) 0 0
\(439\) −3.70145e6 −0.916666 −0.458333 0.888781i \(-0.651553\pi\)
−0.458333 + 0.888781i \(0.651553\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −5.63125e6 −1.36331 −0.681656 0.731673i \(-0.738740\pi\)
−0.681656 + 0.731673i \(0.738740\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −1.59512e6 −0.373404 −0.186702 0.982417i \(-0.559780\pi\)
−0.186702 + 0.982417i \(0.559780\pi\)
\(450\) 0 0
\(451\) −6.62019e6 −1.53260
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −5.83610e6 −1.30717 −0.653585 0.756853i \(-0.726736\pi\)
−0.653585 + 0.756853i \(0.726736\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 4.56280e6 0.999952 0.499976 0.866039i \(-0.333342\pi\)
0.499976 + 0.866039i \(0.333342\pi\)
\(462\) 0 0
\(463\) −5.16917e6 −1.12064 −0.560322 0.828275i \(-0.689323\pi\)
−0.560322 + 0.828275i \(0.689323\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 5.88402e6 1.24848 0.624240 0.781232i \(-0.285409\pi\)
0.624240 + 0.781232i \(0.285409\pi\)
\(468\) 0 0
\(469\) 1.39401e6 0.292639
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2.01792e6 −0.414716
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 8.12131e6 1.61729 0.808644 0.588298i \(-0.200202\pi\)
0.808644 + 0.588298i \(0.200202\pi\)
\(480\) 0 0
\(481\) −3.78772e6 −0.746476
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −4.97424e6 −0.950396 −0.475198 0.879879i \(-0.657624\pi\)
−0.475198 + 0.879879i \(0.657624\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7.77646e6 1.45572 0.727860 0.685725i \(-0.240515\pi\)
0.727860 + 0.685725i \(0.240515\pi\)
\(492\) 0 0
\(493\) −2.09920e6 −0.388988
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.04091e6 −0.189026
\(498\) 0 0
\(499\) −3.28334e6 −0.590290 −0.295145 0.955453i \(-0.595368\pi\)
−0.295145 + 0.955453i \(0.595368\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −2.87651e6 −0.506928 −0.253464 0.967345i \(-0.581570\pi\)
−0.253464 + 0.967345i \(0.581570\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 7.56707e6 1.29459 0.647296 0.762239i \(-0.275900\pi\)
0.647296 + 0.762239i \(0.275900\pi\)
\(510\) 0 0
\(511\) −3.61420e6 −0.612293
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 3.07490e6 0.505946
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −7.07463e6 −1.14185 −0.570926 0.821002i \(-0.693416\pi\)
−0.570926 + 0.821002i \(0.693416\pi\)
\(522\) 0 0
\(523\) 2.61323e6 0.417756 0.208878 0.977942i \(-0.433019\pi\)
0.208878 + 0.977942i \(0.433019\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.70977e7 −2.68170
\(528\) 0 0
\(529\) −5.78259e6 −0.898428
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −7.75830e6 −1.18290
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −318276. −0.0471880
\(540\) 0 0
\(541\) 9.76181e6 1.43396 0.716980 0.697094i \(-0.245524\pi\)
0.716980 + 0.697094i \(0.245524\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −9.10548e6 −1.30117 −0.650586 0.759433i \(-0.725477\pi\)
−0.650586 + 0.759433i \(0.725477\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1.29412e6 −0.181592
\(552\) 0 0
\(553\) 4.20262e6 0.584396
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.70022e6 0.232203 0.116102 0.993237i \(-0.462960\pi\)
0.116102 + 0.993237i \(0.462960\pi\)
\(558\) 0 0
\(559\) −2.36483e6 −0.320088
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 5.46904e6 0.727177 0.363588 0.931560i \(-0.381551\pi\)
0.363588 + 0.931560i \(0.381551\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.43662e7 −1.86020 −0.930101 0.367305i \(-0.880281\pi\)
−0.930101 + 0.367305i \(0.880281\pi\)
\(570\) 0 0
\(571\) −1.53444e6 −0.196952 −0.0984760 0.995139i \(-0.531397\pi\)
−0.0984760 + 0.995139i \(0.531397\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.30952e7 −1.63747 −0.818736 0.574170i \(-0.805325\pi\)
−0.818736 + 0.574170i \(0.805325\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1.16564e6 −0.143260
\(582\) 0 0
\(583\) 1.05501e7 1.28554
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.47314e6 0.416032 0.208016 0.978125i \(-0.433299\pi\)
0.208016 + 0.978125i \(0.433299\pi\)
\(588\) 0 0
\(589\) −1.05404e7 −1.25190
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 9.42933e6 1.10114 0.550572 0.834788i \(-0.314410\pi\)
0.550572 + 0.834788i \(0.314410\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1.54909e7 1.76405 0.882025 0.471203i \(-0.156180\pi\)
0.882025 + 0.471203i \(0.156180\pi\)
\(600\) 0 0
\(601\) 5.05608e6 0.570989 0.285494 0.958380i \(-0.407842\pi\)
0.285494 + 0.958380i \(0.407842\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 548419. 0.0604144 0.0302072 0.999544i \(-0.490383\pi\)
0.0302072 + 0.999544i \(0.490383\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.60351e6 0.390502
\(612\) 0 0
\(613\) 4.93844e6 0.530809 0.265404 0.964137i \(-0.414495\pi\)
0.265404 + 0.964137i \(0.414495\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.92279e6 0.732097 0.366048 0.930596i \(-0.380710\pi\)
0.366048 + 0.930596i \(0.380710\pi\)
\(618\) 0 0
\(619\) −3.07648e6 −0.322721 −0.161361 0.986896i \(-0.551588\pi\)
−0.161361 + 0.986896i \(0.551588\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −2.86807e6 −0.296053
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1.99399e7 −2.00953
\(630\) 0 0
\(631\) −6.34397e6 −0.634290 −0.317145 0.948377i \(-0.602724\pi\)
−0.317145 + 0.948377i \(0.602724\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −372992. −0.0364209
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1.79976e6 −0.173010 −0.0865049 0.996251i \(-0.527570\pi\)
−0.0865049 + 0.996251i \(0.527570\pi\)
\(642\) 0 0
\(643\) −9.17436e6 −0.875081 −0.437541 0.899199i \(-0.644150\pi\)
−0.437541 + 0.899199i \(0.644150\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4.16863e6 0.391501 0.195751 0.980654i \(-0.437286\pi\)
0.195751 + 0.980654i \(0.437286\pi\)
\(648\) 0 0
\(649\) −1.69457e7 −1.57924
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3.18235e6 0.292056 0.146028 0.989280i \(-0.453351\pi\)
0.146028 + 0.989280i \(0.453351\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1.39238e7 −1.24894 −0.624472 0.781047i \(-0.714686\pi\)
−0.624472 + 0.781047i \(0.714686\pi\)
\(660\) 0 0
\(661\) 9.49559e6 0.845314 0.422657 0.906290i \(-0.361097\pi\)
0.422657 + 0.906290i \(0.361097\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −795773. −0.0692587
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 146382. 0.0125511
\(672\) 0 0
\(673\) 1.66532e7 1.41729 0.708647 0.705563i \(-0.249306\pi\)
0.708647 + 0.705563i \(0.249306\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.04609e7 0.877200 0.438600 0.898682i \(-0.355474\pi\)
0.438600 + 0.898682i \(0.355474\pi\)
\(678\) 0 0
\(679\) −7.01634e6 −0.584031
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.75109e7 1.43633 0.718167 0.695870i \(-0.244981\pi\)
0.718167 + 0.695870i \(0.244981\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.23638e7 0.992210
\(690\) 0 0
\(691\) −9.62282e6 −0.766667 −0.383334 0.923610i \(-0.625224\pi\)
−0.383334 + 0.923610i \(0.625224\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −4.08423e7 −3.18440
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −2.07535e7 −1.59513 −0.797566 0.603232i \(-0.793879\pi\)
−0.797566 + 0.603232i \(0.793879\pi\)
\(702\) 0 0
\(703\) −1.22926e7 −0.938115
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.70561e6 0.730255
\(708\) 0 0
\(709\) −1.28077e7 −0.956873 −0.478436 0.878122i \(-0.658796\pi\)
−0.478436 + 0.878122i \(0.658796\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −6.48146e6 −0.477473
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 3.51029e6 0.253233 0.126617 0.991952i \(-0.459588\pi\)
0.126617 + 0.991952i \(0.459588\pi\)
\(720\) 0 0
\(721\) 380825. 0.0272827
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −2.05040e7 −1.43881 −0.719403 0.694593i \(-0.755584\pi\)
−0.719403 + 0.694593i \(0.755584\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1.24492e7 −0.861687
\(732\) 0 0
\(733\) 1.64271e7 1.12928 0.564639 0.825338i \(-0.309015\pi\)
0.564639 + 0.825338i \(0.309015\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −3.61969e6 −0.245472
\(738\) 0 0
\(739\) −1.40579e7 −0.946910 −0.473455 0.880818i \(-0.656993\pi\)
−0.473455 + 0.880818i \(0.656993\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.36451e7 −0.906784 −0.453392 0.891311i \(-0.649786\pi\)
−0.453392 + 0.891311i \(0.649786\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1.61036e7 1.04886
\(750\) 0 0
\(751\) −1.90777e7 −1.23431 −0.617156 0.786841i \(-0.711715\pi\)
−0.617156 + 0.786841i \(0.711715\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.41302e7 −0.896205 −0.448102 0.893982i \(-0.647900\pi\)
−0.448102 + 0.893982i \(0.647900\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 7.86587e6 0.492363 0.246181 0.969224i \(-0.420824\pi\)
0.246181 + 0.969224i \(0.420824\pi\)
\(762\) 0 0
\(763\) 5.47511e6 0.340472
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.98589e7 −1.21890
\(768\) 0 0
\(769\) 1.10128e6 0.0671553 0.0335776 0.999436i \(-0.489310\pi\)
0.0335776 + 0.999436i \(0.489310\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 519333. 0.0312606 0.0156303 0.999878i \(-0.495025\pi\)
0.0156303 + 0.999878i \(0.495025\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2.51786e7 −1.48658
\(780\) 0 0
\(781\) 2.70283e6 0.158559
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 5.27130e6 0.303376 0.151688 0.988428i \(-0.451529\pi\)
0.151688 + 0.988428i \(0.451529\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.84590e7 1.04898
\(792\) 0 0
\(793\) 171547. 0.00968726
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.05709e7 −0.589479 −0.294739 0.955578i \(-0.595233\pi\)
−0.294739 + 0.955578i \(0.595233\pi\)
\(798\) 0 0
\(799\) 1.89701e7 1.05124
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 9.38465e6 0.513605
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.56280e7 0.839522 0.419761 0.907635i \(-0.362114\pi\)
0.419761 + 0.907635i \(0.362114\pi\)
\(810\) 0 0
\(811\) −6.16341e6 −0.329055 −0.164528 0.986372i \(-0.552610\pi\)
−0.164528 + 0.986372i \(0.552610\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −7.67477e6 −0.402263
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 3.11036e7 1.61047 0.805236 0.592955i \(-0.202039\pi\)
0.805236 + 0.592955i \(0.202039\pi\)
\(822\) 0 0
\(823\) −3.53899e6 −0.182129 −0.0910646 0.995845i \(-0.529027\pi\)
−0.0910646 + 0.995845i \(0.529027\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.05076e7 1.04268 0.521341 0.853349i \(-0.325432\pi\)
0.521341 + 0.853349i \(0.325432\pi\)
\(828\) 0 0
\(829\) −1.36626e7 −0.690474 −0.345237 0.938516i \(-0.612201\pi\)
−0.345237 + 0.938516i \(0.612201\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.96356e6 −0.0980462
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 5.83361e6 0.286110 0.143055 0.989715i \(-0.454307\pi\)
0.143055 + 0.989715i \(0.454307\pi\)
\(840\) 0 0
\(841\) −1.95425e7 −0.952775
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −5.52886e6 −0.264805
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −7.55889e6 −0.357795
\(852\) 0 0
\(853\) 1.90084e7 0.894483 0.447241 0.894413i \(-0.352406\pi\)
0.447241 + 0.894413i \(0.352406\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.44094e7 −0.670183 −0.335091 0.942186i \(-0.608767\pi\)
−0.335091 + 0.942186i \(0.608767\pi\)
\(858\) 0 0
\(859\) −2.75606e7 −1.27440 −0.637200 0.770699i \(-0.719907\pi\)
−0.637200 + 0.770699i \(0.719907\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −3.96582e7 −1.81262 −0.906308 0.422617i \(-0.861112\pi\)
−0.906308 + 0.422617i \(0.861112\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −1.09126e7 −0.490204
\(870\) 0 0
\(871\) −4.24196e6 −0.189462
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.26699e7 −0.556257 −0.278129 0.960544i \(-0.589714\pi\)
−0.278129 + 0.960544i \(0.589714\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.66070e7 0.720859 0.360429 0.932786i \(-0.382630\pi\)
0.360429 + 0.932786i \(0.382630\pi\)
\(882\) 0 0
\(883\) −2.01030e7 −0.867679 −0.433839 0.900990i \(-0.642841\pi\)
−0.433839 + 0.900990i \(0.642841\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −2.14814e7 −0.916757 −0.458378 0.888757i \(-0.651570\pi\)
−0.458378 + 0.888757i \(0.651570\pi\)
\(888\) 0 0
\(889\) 3.37968e7 1.43424
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.16948e7 0.490754
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 7.88948e6 0.325574
\(900\) 0 0
\(901\) 6.50872e7 2.67106
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 2.05682e7 0.830191 0.415095 0.909778i \(-0.363748\pi\)
0.415095 + 0.909778i \(0.363748\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −4.99257e6 −0.199310 −0.0996548 0.995022i \(-0.531774\pi\)
−0.0996548 + 0.995022i \(0.531774\pi\)
\(912\) 0 0
\(913\) 3.02671e6 0.120169
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −4.80555e7 −1.88721
\(918\) 0 0
\(919\) −2.66030e7 −1.03906 −0.519531 0.854451i \(-0.673893\pi\)
−0.519531 + 0.854451i \(0.673893\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 3.16749e6 0.122380
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 4.26521e7 1.62144 0.810721 0.585433i \(-0.199076\pi\)
0.810721 + 0.585433i \(0.199076\pi\)
\(930\) 0 0
\(931\) −1.21050e6 −0.0457711
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 4.73064e7 1.76024 0.880118 0.474754i \(-0.157463\pi\)
0.880118 + 0.474754i \(0.157463\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −8.44452e6 −0.310886 −0.155443 0.987845i \(-0.549680\pi\)
−0.155443 + 0.987845i \(0.549680\pi\)
\(942\) 0 0
\(943\) −1.54827e7 −0.566979
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 2.76156e7 1.00064 0.500321 0.865840i \(-0.333215\pi\)
0.500321 + 0.865840i \(0.333215\pi\)
\(948\) 0 0
\(949\) 1.09980e7 0.396413
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −3.91798e7 −1.39743 −0.698716 0.715400i \(-0.746245\pi\)
−0.698716 + 0.715400i \(0.746245\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −384221. −0.0134907
\(960\) 0 0
\(961\) 3.56296e7 1.24452
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 3.64633e7 1.25398 0.626989 0.779028i \(-0.284287\pi\)
0.626989 + 0.779028i \(0.284287\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −5.50829e7 −1.87486 −0.937429 0.348176i \(-0.886801\pi\)
−0.937429 + 0.348176i \(0.886801\pi\)
\(972\) 0 0
\(973\) 5.62404e7 1.90443
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 4.10108e7 1.37455 0.687277 0.726395i \(-0.258806\pi\)
0.687277 + 0.726395i \(0.258806\pi\)
\(978\) 0 0
\(979\) 7.44724e6 0.248335
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 5.35318e7 1.76696 0.883482 0.468465i \(-0.155193\pi\)
0.883482 + 0.468465i \(0.155193\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −4.71931e6 −0.153422
\(990\) 0 0
\(991\) −2.57465e7 −0.832788 −0.416394 0.909184i \(-0.636706\pi\)
−0.416394 + 0.909184i \(0.636706\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −556486. −0.0177303 −0.00886515 0.999961i \(-0.502822\pi\)
−0.00886515 + 0.999961i \(0.502822\pi\)
\(998\) 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 900.6.a.u.1.2 2
3.2 odd 2 900.6.a.t.1.2 2
5.2 odd 4 900.6.d.n.649.4 4
5.3 odd 4 900.6.d.n.649.1 4
5.4 even 2 180.6.a.f.1.1 2
15.2 even 4 900.6.d.k.649.4 4
15.8 even 4 900.6.d.k.649.1 4
15.14 odd 2 180.6.a.g.1.1 yes 2
20.19 odd 2 720.6.a.bc.1.2 2
60.59 even 2 720.6.a.bg.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.6.a.f.1.1 2 5.4 even 2
180.6.a.g.1.1 yes 2 15.14 odd 2
720.6.a.bc.1.2 2 20.19 odd 2
720.6.a.bg.1.2 2 60.59 even 2
900.6.a.t.1.2 2 3.2 odd 2
900.6.a.u.1.2 2 1.1 even 1 trivial
900.6.d.k.649.1 4 15.8 even 4
900.6.d.k.649.4 4 15.2 even 4
900.6.d.n.649.1 4 5.3 odd 4
900.6.d.n.649.4 4 5.2 odd 4