Properties

Label 1800.2.b.f.251.3
Level $1800$
Weight $2$
Character 1800.251
Analytic conductor $14.373$
Analytic rank $0$
Dimension $8$
CM discriminant -40
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1800,2,Mod(251,1800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1800.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1800.b (of order \(2\), degree \(1\), 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: \(14.3730723638\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 251.3
Root \(-1.14412 + 1.14412i\) of defining polynomial
Character \(\chi\) \(=\) 1800.251
Dual form 1800.2.b.f.251.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421 q^{2} +2.00000 q^{4} +1.16228i q^{7} -2.82843 q^{8} +O(q^{10})\) \(q-1.41421 q^{2} +2.00000 q^{4} +1.16228i q^{7} -2.82843 q^{8} +5.88635i q^{11} -7.16228i q^{13} -1.64371i q^{14} +4.00000 q^{16} -6.32456 q^{19} -8.32456i q^{22} -4.47214 q^{23} +10.1290i q^{26} +2.32456i q^{28} -5.65685 q^{32} -4.83772i q^{37} +8.94427 q^{38} -7.53006i q^{41} +11.7727i q^{44} +6.32456 q^{46} -2.82843 q^{47} +5.64911 q^{49} -14.3246i q^{52} -5.65685 q^{53} -3.28742i q^{56} -14.3716i q^{59} +8.00000 q^{64} +6.84157i q^{74} -12.6491 q^{76} -6.84157 q^{77} +10.6491i q^{82} -16.6491i q^{88} -0.955223i q^{89} +8.32456 q^{91} -8.94427 q^{92} +4.00000 q^{94} -7.98905 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{4} + 32 q^{16} - 56 q^{49} + 64 q^{64} + 16 q^{91} + 32 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 −1.00000
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.16228i 0.439300i 0.975579 + 0.219650i \(0.0704915\pi\)
−0.975579 + 0.219650i \(0.929509\pi\)
\(8\) −2.82843 −1.00000
\(9\) 0 0
\(10\) 0 0
\(11\) 5.88635i 1.77480i 0.460999 + 0.887401i \(0.347491\pi\)
−0.460999 + 0.887401i \(0.652509\pi\)
\(12\) 0 0
\(13\) − 7.16228i − 1.98646i −0.116171 0.993229i \(-0.537062\pi\)
0.116171 0.993229i \(-0.462938\pi\)
\(14\) − 1.64371i − 0.439300i
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −6.32456 −1.45095 −0.725476 0.688247i \(-0.758380\pi\)
−0.725476 + 0.688247i \(0.758380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 8.32456i − 1.77480i
\(23\) −4.47214 −0.932505 −0.466252 0.884652i \(-0.654396\pi\)
−0.466252 + 0.884652i \(0.654396\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 10.1290i 1.98646i
\(27\) 0 0
\(28\) 2.32456i 0.439300i
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −5.65685 −1.00000
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 4.83772i − 0.795317i −0.917534 0.397658i \(-0.869823\pi\)
0.917534 0.397658i \(-0.130177\pi\)
\(38\) 8.94427 1.45095
\(39\) 0 0
\(40\) 0 0
\(41\) − 7.53006i − 1.17600i −0.808862 0.587999i \(-0.799916\pi\)
0.808862 0.587999i \(-0.200084\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 11.7727i 1.77480i
\(45\) 0 0
\(46\) 6.32456 0.932505
\(47\) −2.82843 −0.412568 −0.206284 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 0 0
\(49\) 5.64911 0.807016
\(50\) 0 0
\(51\) 0 0
\(52\) − 14.3246i − 1.98646i
\(53\) −5.65685 −0.777029 −0.388514 0.921443i \(-0.627012\pi\)
−0.388514 + 0.921443i \(0.627012\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) − 3.28742i − 0.439300i
\(57\) 0 0
\(58\) 0 0
\(59\) − 14.3716i − 1.87103i −0.353291 0.935513i \(-0.614937\pi\)
0.353291 0.935513i \(-0.385063\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 6.84157i 0.795317i
\(75\) 0 0
\(76\) −12.6491 −1.45095
\(77\) −6.84157 −0.779670
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 10.6491i 1.17600i
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) − 16.6491i − 1.77480i
\(89\) − 0.955223i − 0.101253i −0.998718 0.0506267i \(-0.983878\pi\)
0.998718 0.0506267i \(-0.0161219\pi\)
\(90\) 0 0
\(91\) 8.32456 0.872651
\(92\) −8.94427 −0.932505
\(93\) 0 0
\(94\) 4.00000 0.412568
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) −7.98905 −0.807016
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 10.8377i 1.06787i 0.845525 + 0.533936i \(0.179288\pi\)
−0.845525 + 0.533936i \(0.820712\pi\)
\(104\) 20.2580i 1.98646i
\(105\) 0 0
\(106\) 8.00000 0.777029
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.64911i 0.439300i
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 20.3246i 1.87103i
\(119\) 0 0
\(120\) 0 0
\(121\) −23.6491 −2.14992
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 17.8114i − 1.58051i −0.612781 0.790253i \(-0.709949\pi\)
0.612781 0.790253i \(-0.290051\pi\)
\(128\) −11.3137 −1.00000
\(129\) 0 0
\(130\) 0 0
\(131\) − 11.0842i − 0.968432i −0.874948 0.484216i \(-0.839105\pi\)
0.874948 0.484216i \(-0.160895\pi\)
\(132\) 0 0
\(133\) − 7.35089i − 0.637403i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 42.1597 3.52557
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) − 9.67544i − 0.795317i
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 17.8885 1.45095
\(153\) 0 0
\(154\) 9.67544 0.779670
\(155\) 0 0
\(156\) 0 0
\(157\) − 23.8114i − 1.90036i −0.311708 0.950178i \(-0.600901\pi\)
0.311708 0.950178i \(-0.399099\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 5.19786i − 0.409649i
\(162\) 0 0
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) − 15.0601i − 1.17600i
\(165\) 0 0
\(166\) 0 0
\(167\) 4.47214 0.346064 0.173032 0.984916i \(-0.444644\pi\)
0.173032 + 0.984916i \(0.444644\pi\)
\(168\) 0 0
\(169\) −38.2982 −2.94602
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −22.6274 −1.72033 −0.860165 0.510015i \(-0.829640\pi\)
−0.860165 + 0.510015i \(0.829640\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 23.5454i 1.77480i
\(177\) 0 0
\(178\) 1.35089i 0.101253i
\(179\) − 22.8569i − 1.70841i −0.519940 0.854203i \(-0.674046\pi\)
0.519940 0.854203i \(-0.325954\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) −11.7727 −0.872651
\(183\) 0 0
\(184\) 12.6491 0.932505
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −5.65685 −0.412568
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 11.2982 0.807016
\(197\) −22.3607 −1.59313 −0.796566 0.604551i \(-0.793352\pi\)
−0.796566 + 0.604551i \(0.793352\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) − 15.3269i − 1.06787i
\(207\) 0 0
\(208\) − 28.6491i − 1.98646i
\(209\) − 37.2285i − 2.57515i
\(210\) 0 0
\(211\) −22.0000 −1.51454 −0.757271 0.653101i \(-0.773468\pi\)
−0.757271 + 0.653101i \(0.773468\pi\)
\(212\) −11.3137 −0.777029
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 29.8114i 1.99632i 0.0606498 + 0.998159i \(0.480683\pi\)
−0.0606498 + 0.998159i \(0.519317\pi\)
\(224\) − 6.57484i − 0.439300i
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) − 28.7433i − 1.87103i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 25.2982 1.62960 0.814801 0.579741i \(-0.196846\pi\)
0.814801 + 0.579741i \(0.196846\pi\)
\(242\) 33.4449 2.14992
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 45.2982i 2.88226i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 20.9465i − 1.32213i −0.750329 0.661065i \(-0.770105\pi\)
0.750329 0.661065i \(-0.229895\pi\)
\(252\) 0 0
\(253\) − 26.3246i − 1.65501i
\(254\) 25.1891i 1.58051i
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 5.62278 0.349382
\(260\) 0 0
\(261\) 0 0
\(262\) 15.6754i 0.968432i
\(263\) −31.3050 −1.93035 −0.965173 0.261612i \(-0.915746\pi\)
−0.965173 + 0.261612i \(0.915746\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 10.3957i 0.637403i
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 14.1359i 0.849347i 0.905347 + 0.424673i \(0.139611\pi\)
−0.905347 + 0.424673i \(0.860389\pi\)
\(278\) 19.7990 1.18746
\(279\) 0 0
\(280\) 0 0
\(281\) 2.33219i 0.139127i 0.997578 + 0.0695635i \(0.0221607\pi\)
−0.997578 + 0.0695635i \(0.977839\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −59.6228 −3.52557
\(287\) 8.75202 0.516615
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −4.47214 −0.261265 −0.130632 0.991431i \(-0.541701\pi\)
−0.130632 + 0.991431i \(0.541701\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 13.6831i 0.795317i
\(297\) 0 0
\(298\) 0 0
\(299\) 32.0307i 1.85238i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −25.2982 −1.45095
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) −13.6831 −0.779670
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 33.6744i 1.90036i
\(315\) 0 0
\(316\) 0 0
\(317\) 31.3050 1.75826 0.879131 0.476581i \(-0.158124\pi\)
0.879131 + 0.476581i \(0.158124\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 7.35089i 0.409649i
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 21.2982i 1.17600i
\(329\) − 3.28742i − 0.181241i
\(330\) 0 0
\(331\) −31.6228 −1.73814 −0.869072 0.494685i \(-0.835284\pi\)
−0.869072 + 0.494685i \(0.835284\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −6.32456 −0.346064
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) 54.1619 2.94602
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 14.7018i 0.793821i
\(344\) 0 0
\(345\) 0 0
\(346\) 32.0000 1.72033
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 33.2982i − 1.77480i
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) − 1.91045i − 0.101253i
\(357\) 0 0
\(358\) 32.3246i 1.70841i
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 21.0000 1.10526
\(362\) 0 0
\(363\) 0 0
\(364\) 16.6491 0.872651
\(365\) 0 0
\(366\) 0 0
\(367\) 6.18861i 0.323043i 0.986869 + 0.161521i \(0.0516401\pi\)
−0.986869 + 0.161521i \(0.948360\pi\)
\(368\) −17.8885 −0.932505
\(369\) 0 0
\(370\) 0 0
\(371\) − 6.57484i − 0.341348i
\(372\) 0 0
\(373\) − 38.1359i − 1.97460i −0.158854 0.987302i \(-0.550780\pi\)
0.158854 0.987302i \(-0.449220\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) 0 0
\(378\) 0 0
\(379\) 34.0000 1.74646 0.873231 0.487306i \(-0.162020\pi\)
0.873231 + 0.487306i \(0.162020\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 36.7696 1.87884 0.939418 0.342773i \(-0.111366\pi\)
0.939418 + 0.342773i \(0.111366\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −15.9781 −0.807016
\(393\) 0 0
\(394\) 31.6228 1.59313
\(395\) 0 0
\(396\) 0 0
\(397\) 31.1623i 1.56399i 0.623285 + 0.781995i \(0.285798\pi\)
−0.623285 + 0.781995i \(0.714202\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 17.9258i 0.895171i 0.894241 + 0.447586i \(0.147716\pi\)
−0.894241 + 0.447586i \(0.852284\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 28.4765 1.41153
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 21.6754i 1.06787i
\(413\) 16.7038 0.821942
\(414\) 0 0
\(415\) 0 0
\(416\) 40.5160i 1.98646i
\(417\) 0 0
\(418\) 52.6491i 2.57515i
\(419\) 3.97590i 0.194236i 0.995273 + 0.0971178i \(0.0309624\pi\)
−0.995273 + 0.0971178i \(0.969038\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 31.1127 1.51454
\(423\) 0 0
\(424\) 16.0000 0.777029
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 28.2843 1.35302
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) − 42.1597i − 1.99632i
\(447\) 0 0
\(448\) 9.29822i 0.439300i
\(449\) 6.15309i 0.290382i 0.989404 + 0.145191i \(0.0463797\pi\)
−0.989404 + 0.145191i \(0.953620\pi\)
\(450\) 0 0
\(451\) 44.3246 2.08716
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 41.8114i 1.94314i 0.236752 + 0.971570i \(0.423917\pi\)
−0.236752 + 0.971570i \(0.576083\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 40.6491i 1.87103i
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −34.6491 −1.57986
\(482\) −35.7771 −1.62960
\(483\) 0 0
\(484\) −47.2982 −2.14992
\(485\) 0 0
\(486\) 0 0
\(487\) 44.1359i 1.99999i 0.00308010 + 0.999995i \(0.499020\pi\)
−0.00308010 + 0.999995i \(0.500980\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 32.7192i 1.47660i 0.674475 + 0.738298i \(0.264370\pi\)
−0.674475 + 0.738298i \(0.735630\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) − 64.0614i − 2.88226i
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −44.2719 −1.98188 −0.990941 0.134298i \(-0.957122\pi\)
−0.990941 + 0.134298i \(0.957122\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 29.6228i 1.32213i
\(503\) 19.7990 0.882793 0.441397 0.897312i \(-0.354483\pi\)
0.441397 + 0.897312i \(0.354483\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 37.2285i 1.65501i
\(507\) 0 0
\(508\) − 35.6228i − 1.58051i
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −22.6274 −1.00000
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 16.6491i − 0.732227i
\(518\) −7.95181 −0.349382
\(519\) 0 0
\(520\) 0 0
\(521\) 44.7586i 1.96091i 0.196744 + 0.980455i \(0.436963\pi\)
−0.196744 + 0.980455i \(0.563037\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) − 22.1684i − 0.968432i
\(525\) 0 0
\(526\) 44.2719 1.93035
\(527\) 0 0
\(528\) 0 0
\(529\) −3.00000 −0.130435
\(530\) 0 0
\(531\) 0 0
\(532\) − 14.7018i − 0.637403i
\(533\) −53.9324 −2.33607
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 33.2526i 1.43229i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) − 19.9912i − 0.849347i
\(555\) 0 0
\(556\) −28.0000 −1.18746
\(557\) −45.2548 −1.91751 −0.958754 0.284236i \(-0.908260\pi\)
−0.958754 + 0.284236i \(0.908260\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) − 3.29822i − 0.139127i
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 41.4712i 1.73856i 0.494318 + 0.869281i \(0.335418\pi\)
−0.494318 + 0.869281i \(0.664582\pi\)
\(570\) 0 0
\(571\) 44.2719 1.85272 0.926360 0.376638i \(-0.122920\pi\)
0.926360 + 0.376638i \(0.122920\pi\)
\(572\) 84.3193 3.52557
\(573\) 0 0
\(574\) −12.3772 −0.516615
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) −24.0416 −1.00000
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) − 33.2982i − 1.37907i
\(584\) 0 0
\(585\) 0 0
\(586\) 6.32456 0.261265
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) − 19.3509i − 0.795317i
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) − 45.2982i − 1.85238i
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 25.2982 1.03194 0.515968 0.856608i \(-0.327432\pi\)
0.515968 + 0.856608i \(0.327432\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 36.7851i − 1.49306i −0.665352 0.746530i \(-0.731719\pi\)
0.665352 0.746530i \(-0.268281\pi\)
\(608\) 35.7771 1.45095
\(609\) 0 0
\(610\) 0 0
\(611\) 20.2580i 0.819550i
\(612\) 0 0
\(613\) 30.7851i 1.24340i 0.783257 + 0.621698i \(0.213557\pi\)
−0.783257 + 0.621698i \(0.786443\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 19.3509 0.779670
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 46.0000 1.84890 0.924448 0.381308i \(-0.124526\pi\)
0.924448 + 0.381308i \(0.124526\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1.11023 0.0444806
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) − 47.6228i − 1.90036i
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −44.2719 −1.75826
\(635\) 0 0
\(636\) 0 0
\(637\) − 40.4605i − 1.60310i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 34.3629i − 1.35725i −0.734484 0.678626i \(-0.762576\pi\)
0.734484 0.678626i \(-0.237424\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) − 10.3957i − 0.409649i
\(645\) 0 0
\(646\) 0 0
\(647\) 31.1127 1.22317 0.611583 0.791180i \(-0.290533\pi\)
0.611583 + 0.791180i \(0.290533\pi\)
\(648\) 0 0
\(649\) 84.5964 3.32070
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −4.47214 −0.175008 −0.0875041 0.996164i \(-0.527889\pi\)
−0.0875041 + 0.996164i \(0.527889\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) − 30.1202i − 1.17600i
\(657\) 0 0
\(658\) 4.64911i 0.181241i
\(659\) − 49.6897i − 1.93564i −0.251649 0.967818i \(-0.580973\pi\)
0.251649 0.967818i \(-0.419027\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 44.7214 1.73814
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 8.94427 0.346064
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −76.5964 −2.94602
\(677\) −49.1935 −1.89066 −0.945330 0.326116i \(-0.894260\pi\)
−0.945330 + 0.326116i \(0.894260\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) − 20.7915i − 0.793821i
\(687\) 0 0
\(688\) 0 0
\(689\) 40.5160i 1.54354i
\(690\) 0 0
\(691\) −31.6228 −1.20299 −0.601494 0.798878i \(-0.705427\pi\)
−0.601494 + 0.798878i \(0.705427\pi\)
\(692\) −45.2548 −1.72033
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 30.5964i 1.15397i
\(704\) 47.0908i 1.77480i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 2.70178i 0.101253i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) − 45.7138i − 1.70841i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −12.5964 −0.469116
\(722\) −29.6985 −1.10526
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 34.8377i − 1.29206i −0.763312 0.646030i \(-0.776428\pi\)
0.763312 0.646030i \(-0.223572\pi\)
\(728\) −23.5454 −0.872651
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 18.7851i 0.693842i 0.937894 + 0.346921i \(0.112773\pi\)
−0.937894 + 0.346921i \(0.887227\pi\)
\(734\) − 8.75202i − 0.323043i
\(735\) 0 0
\(736\) 25.2982 0.932505
\(737\) 0 0
\(738\) 0 0
\(739\) −6.32456 −0.232653 −0.116326 0.993211i \(-0.537112\pi\)
−0.116326 + 0.993211i \(0.537112\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 9.29822i 0.341348i
\(743\) 36.7696 1.34894 0.674472 0.738300i \(-0.264371\pi\)
0.674472 + 0.738300i \(0.264371\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 53.9324i 1.97460i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) −11.3137 −0.412568
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 9.86406i − 0.358515i −0.983802 0.179258i \(-0.942630\pi\)
0.983802 0.179258i \(-0.0573696\pi\)
\(758\) −48.0833 −1.74646
\(759\) 0 0
\(760\) 0 0
\(761\) − 51.3334i − 1.86084i −0.366501 0.930418i \(-0.619444\pi\)
0.366501 0.930418i \(-0.380556\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) −52.0000 −1.87884
\(767\) −102.934 −3.71672
\(768\) 0 0
\(769\) 12.6491 0.456139 0.228069 0.973645i \(-0.426759\pi\)
0.228069 + 0.973645i \(0.426759\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 22.3607 0.804258 0.402129 0.915583i \(-0.368270\pi\)
0.402129 + 0.915583i \(0.368270\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 47.6243i 1.70632i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 22.5964 0.807016
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) −44.7214 −1.59313
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) − 44.0701i − 1.56399i
\(795\) 0 0
\(796\) 0 0
\(797\) 39.5980 1.40263 0.701316 0.712850i \(-0.252596\pi\)
0.701316 + 0.712850i \(0.252596\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) − 25.3509i − 0.895171i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 17.3923i 0.611481i 0.952115 + 0.305741i \(0.0989040\pi\)
−0.952115 + 0.305741i \(0.901096\pi\)
\(810\) 0 0
\(811\) 2.00000 0.0702295 0.0351147 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −40.2719 −1.41153
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 19.7990 0.692255
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) − 56.1359i − 1.95678i −0.206778 0.978388i \(-0.566298\pi\)
0.206778 0.978388i \(-0.433702\pi\)
\(824\) − 30.6537i − 1.06787i
\(825\) 0 0
\(826\) −23.6228 −0.821942
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 57.2982i − 1.98646i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) − 74.4571i − 2.57515i
\(837\) 0 0
\(838\) − 5.62278i − 0.194236i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −44.0000 −1.51454
\(845\) 0 0
\(846\) 0 0
\(847\) − 27.4868i − 0.944459i
\(848\) −22.6274 −0.777029
\(849\) 0 0
\(850\) 0 0
\(851\) 21.6350i 0.741637i
\(852\) 0 0
\(853\) − 45.1096i − 1.54452i −0.635304 0.772262i \(-0.719125\pi\)
0.635304 0.772262i \(-0.280875\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −14.0000 −0.477674 −0.238837 0.971060i \(-0.576766\pi\)
−0.238837 + 0.971060i \(0.576766\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −58.1378 −1.97903 −0.989516 0.144421i \(-0.953868\pi\)
−0.989516 + 0.144421i \(0.953868\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) −40.0000 −1.35302
\(875\) 0 0
\(876\) 0 0
\(877\) 33.1096i 1.11803i 0.829157 + 0.559016i \(0.188821\pi\)
−0.829157 + 0.559016i \(0.811179\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 49.9565i − 1.68308i −0.540198 0.841538i \(-0.681651\pi\)
0.540198 0.841538i \(-0.318349\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 4.47214 0.150160 0.0750798 0.997178i \(-0.476079\pi\)
0.0750798 + 0.997178i \(0.476079\pi\)
\(888\) 0 0
\(889\) 20.7018 0.694315
\(890\) 0 0
\(891\) 0 0
\(892\) 59.6228i 1.99632i
\(893\) 17.8885 0.598617
\(894\) 0 0
\(895\) 0 0
\(896\) − 13.1497i − 0.439300i
\(897\) 0 0
\(898\) − 8.70178i − 0.290382i
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) −62.6844 −2.08716
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 12.8829 0.425432
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) − 59.1302i − 1.94314i
\(927\) 0 0
\(928\) 0 0
\(929\) 59.8187i 1.96259i 0.192514 + 0.981294i \(0.438336\pi\)
−0.192514 + 0.981294i \(0.561664\pi\)
\(930\) 0 0
\(931\) −35.7281 −1.17094
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 33.6754i 1.09662i
\(944\) − 57.4865i − 1.87103i
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 31.0000 1.00000
\(962\) 49.0012 1.57986
\(963\) 0 0
\(964\) 50.5964 1.62960
\(965\) 0 0
\(966\) 0 0
\(967\) − 15.8641i − 0.510154i −0.966921 0.255077i \(-0.917899\pi\)
0.966921 0.255077i \(-0.0821008\pi\)
\(968\) 66.8898 2.14992
\(969\) 0 0
\(970\) 0 0
\(971\) 48.3128i 1.55043i 0.631697 + 0.775215i \(0.282359\pi\)
−0.631697 + 0.775215i \(0.717641\pi\)
\(972\) 0 0
\(973\) − 16.2719i − 0.521653i
\(974\) − 62.4177i − 1.99999i
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 5.62278 0.179705
\(980\) 0 0
\(981\) 0 0
\(982\) − 46.2719i − 1.47660i
\(983\) −48.0833 −1.53362 −0.766809 0.641875i \(-0.778157\pi\)
−0.766809 + 0.641875i \(0.778157\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 90.5964i 2.88226i
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 6.78505i − 0.214885i −0.994211 0.107442i \(-0.965734\pi\)
0.994211 0.107442i \(-0.0342661\pi\)
\(998\) 62.6099 1.98188
\(999\) 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 1800.2.b.f.251.3 8
3.2 odd 2 inner 1800.2.b.f.251.7 8
4.3 odd 2 7200.2.b.f.4751.3 8
5.2 odd 4 360.2.m.b.179.2 yes 4
5.3 odd 4 360.2.m.a.179.3 yes 4
5.4 even 2 inner 1800.2.b.f.251.6 8
8.3 odd 2 inner 1800.2.b.f.251.6 8
8.5 even 2 7200.2.b.f.4751.5 8
12.11 even 2 7200.2.b.f.4751.4 8
15.2 even 4 360.2.m.b.179.3 yes 4
15.8 even 4 360.2.m.a.179.2 4
15.14 odd 2 inner 1800.2.b.f.251.2 8
20.3 even 4 1440.2.m.b.719.1 4
20.7 even 4 1440.2.m.a.719.4 4
20.19 odd 2 7200.2.b.f.4751.5 8
24.5 odd 2 7200.2.b.f.4751.6 8
24.11 even 2 inner 1800.2.b.f.251.2 8
40.3 even 4 360.2.m.b.179.2 yes 4
40.13 odd 4 1440.2.m.a.719.4 4
40.19 odd 2 CM 1800.2.b.f.251.3 8
40.27 even 4 360.2.m.a.179.3 yes 4
40.29 even 2 7200.2.b.f.4751.3 8
40.37 odd 4 1440.2.m.b.719.1 4
60.23 odd 4 1440.2.m.b.719.3 4
60.47 odd 4 1440.2.m.a.719.2 4
60.59 even 2 7200.2.b.f.4751.6 8
120.29 odd 2 7200.2.b.f.4751.4 8
120.53 even 4 1440.2.m.a.719.2 4
120.59 even 2 inner 1800.2.b.f.251.7 8
120.77 even 4 1440.2.m.b.719.3 4
120.83 odd 4 360.2.m.b.179.3 yes 4
120.107 odd 4 360.2.m.a.179.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.m.a.179.2 4 15.8 even 4
360.2.m.a.179.2 4 120.107 odd 4
360.2.m.a.179.3 yes 4 5.3 odd 4
360.2.m.a.179.3 yes 4 40.27 even 4
360.2.m.b.179.2 yes 4 5.2 odd 4
360.2.m.b.179.2 yes 4 40.3 even 4
360.2.m.b.179.3 yes 4 15.2 even 4
360.2.m.b.179.3 yes 4 120.83 odd 4
1440.2.m.a.719.2 4 60.47 odd 4
1440.2.m.a.719.2 4 120.53 even 4
1440.2.m.a.719.4 4 20.7 even 4
1440.2.m.a.719.4 4 40.13 odd 4
1440.2.m.b.719.1 4 20.3 even 4
1440.2.m.b.719.1 4 40.37 odd 4
1440.2.m.b.719.3 4 60.23 odd 4
1440.2.m.b.719.3 4 120.77 even 4
1800.2.b.f.251.2 8 15.14 odd 2 inner
1800.2.b.f.251.2 8 24.11 even 2 inner
1800.2.b.f.251.3 8 1.1 even 1 trivial
1800.2.b.f.251.3 8 40.19 odd 2 CM
1800.2.b.f.251.6 8 5.4 even 2 inner
1800.2.b.f.251.6 8 8.3 odd 2 inner
1800.2.b.f.251.7 8 3.2 odd 2 inner
1800.2.b.f.251.7 8 120.59 even 2 inner
7200.2.b.f.4751.3 8 4.3 odd 2
7200.2.b.f.4751.3 8 40.29 even 2
7200.2.b.f.4751.4 8 12.11 even 2
7200.2.b.f.4751.4 8 120.29 odd 2
7200.2.b.f.4751.5 8 8.5 even 2
7200.2.b.f.4751.5 8 20.19 odd 2
7200.2.b.f.4751.6 8 24.5 odd 2
7200.2.b.f.4751.6 8 60.59 even 2