Properties

Label 360.2.d.d.109.3
Level $360$
Weight $2$
Character 360.109
Analytic conductor $2.875$
Analytic rank $0$
Dimension $4$
CM discriminant -24
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [360,2,Mod(109,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.d (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: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 109.3
Root \(0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 360.109
Dual form 360.2.d.d.109.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421 q^{2} +2.00000 q^{4} +(-1.41421 - 1.73205i) q^{5} -4.89898i q^{7} +2.82843 q^{8} +O(q^{10})\) \(q+1.41421 q^{2} +2.00000 q^{4} +(-1.41421 - 1.73205i) q^{5} -4.89898i q^{7} +2.82843 q^{8} +(-2.00000 - 2.44949i) q^{10} +3.46410i q^{11} -6.92820i q^{14} +4.00000 q^{16} +(-2.82843 - 3.46410i) q^{20} +4.89898i q^{22} +(-1.00000 + 4.89898i) q^{25} -9.79796i q^{28} +10.3923i q^{29} +10.0000 q^{31} +5.65685 q^{32} +(-8.48528 + 6.92820i) q^{35} +(-4.00000 - 4.89898i) q^{40} +6.92820i q^{44} -17.0000 q^{49} +(-1.41421 + 6.92820i) q^{50} -14.1421 q^{53} +(6.00000 - 4.89898i) q^{55} -13.8564i q^{56} +14.6969i q^{58} -10.3923i q^{59} +14.1421 q^{62} +8.00000 q^{64} +(-12.0000 + 9.79796i) q^{70} +9.79796i q^{73} +16.9706 q^{77} -10.0000 q^{79} +(-5.65685 - 6.92820i) q^{80} +5.65685 q^{83} +9.79796i q^{88} +19.5959i q^{97} -24.0416 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} - 8 q^{10} + 16 q^{16} - 4 q^{25} + 40 q^{31} - 16 q^{40} - 68 q^{49} + 24 q^{55} + 32 q^{64} - 48 q^{70} - 40 q^{79}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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
\(5\) −1.41421 1.73205i −0.632456 0.774597i
\(6\) 0 0
\(7\) 4.89898i 1.85164i −0.377964 0.925820i \(-0.623376\pi\)
0.377964 0.925820i \(-0.376624\pi\)
\(8\) 2.82843 1.00000
\(9\) 0 0
\(10\) −2.00000 2.44949i −0.632456 0.774597i
\(11\) 3.46410i 1.04447i 0.852803 + 0.522233i \(0.174901\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 6.92820i 1.85164i
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −2.82843 3.46410i −0.632456 0.774597i
\(21\) 0 0
\(22\) 4.89898i 1.04447i
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −1.00000 + 4.89898i −0.200000 + 0.979796i
\(26\) 0 0
\(27\) 0 0
\(28\) 9.79796i 1.85164i
\(29\) 10.3923i 1.92980i 0.262613 + 0.964901i \(0.415416\pi\)
−0.262613 + 0.964901i \(0.584584\pi\)
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 5.65685 1.00000
\(33\) 0 0
\(34\) 0 0
\(35\) −8.48528 + 6.92820i −1.43427 + 1.17108i
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −4.00000 4.89898i −0.632456 0.774597i
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 6.92820i 1.04447i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −17.0000 −2.42857
\(50\) −1.41421 + 6.92820i −0.200000 + 0.979796i
\(51\) 0 0
\(52\) 0 0
\(53\) −14.1421 −1.94257 −0.971286 0.237915i \(-0.923536\pi\)
−0.971286 + 0.237915i \(0.923536\pi\)
\(54\) 0 0
\(55\) 6.00000 4.89898i 0.809040 0.660578i
\(56\) 13.8564i 1.85164i
\(57\) 0 0
\(58\) 14.6969i 1.92980i
\(59\) 10.3923i 1.35296i −0.736460 0.676481i \(-0.763504\pi\)
0.736460 0.676481i \(-0.236496\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 14.1421 1.79605
\(63\) 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
\(70\) −12.0000 + 9.79796i −1.43427 + 1.17108i
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 9.79796i 1.14676i 0.819288 + 0.573382i \(0.194369\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 16.9706 1.93398
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) −5.65685 6.92820i −0.632456 0.774597i
\(81\) 0 0
\(82\) 0 0
\(83\) 5.65685 0.620920 0.310460 0.950586i \(-0.399517\pi\)
0.310460 + 0.950586i \(0.399517\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 9.79796i 1.04447i
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 19.5959i 1.98966i 0.101535 + 0.994832i \(0.467625\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −24.0416 −2.42857
\(99\) 0 0
\(100\) −2.00000 + 9.79796i −0.200000 + 0.979796i
\(101\) 3.46410i 0.344691i −0.985037 0.172345i \(-0.944865\pi\)
0.985037 0.172345i \(-0.0551346\pi\)
\(102\) 0 0
\(103\) 14.6969i 1.44813i −0.689730 0.724066i \(-0.742271\pi\)
0.689730 0.724066i \(-0.257729\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −20.0000 −1.94257
\(107\) 11.3137 1.09374 0.546869 0.837218i \(-0.315820\pi\)
0.546869 + 0.837218i \(0.315820\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 8.48528 6.92820i 0.809040 0.660578i
\(111\) 0 0
\(112\) 19.5959i 1.85164i
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 20.7846i 1.92980i
\(117\) 0 0
\(118\) 14.6969i 1.35296i
\(119\) 0 0
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 20.0000 1.79605
\(125\) 9.89949 5.19615i 0.885438 0.464758i
\(126\) 0 0
\(127\) 4.89898i 0.434714i 0.976092 + 0.217357i \(0.0697436\pi\)
−0.976092 + 0.217357i \(0.930256\pi\)
\(128\) 11.3137 1.00000
\(129\) 0 0
\(130\) 0 0
\(131\) 3.46410i 0.302660i −0.988483 0.151330i \(-0.951644\pi\)
0.988483 0.151330i \(-0.0483556\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) −16.9706 + 13.8564i −1.43427 + 1.17108i
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 18.0000 14.6969i 1.49482 1.22051i
\(146\) 13.8564i 1.14676i
\(147\) 0 0
\(148\) 0 0
\(149\) 24.2487i 1.98653i −0.115857 0.993266i \(-0.536961\pi\)
0.115857 0.993266i \(-0.463039\pi\)
\(150\) 0 0
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 24.0000 1.93398
\(155\) −14.1421 17.3205i −1.13592 1.39122i
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) −14.1421 −1.12509
\(159\) 0 0
\(160\) −8.00000 9.79796i −0.632456 0.774597i
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −19.7990 −1.50529 −0.752645 0.658427i \(-0.771222\pi\)
−0.752645 + 0.658427i \(0.771222\pi\)
\(174\) 0 0
\(175\) 24.0000 + 4.89898i 1.81423 + 0.370328i
\(176\) 13.8564i 1.04447i
\(177\) 0 0
\(178\) 0 0
\(179\) 24.2487i 1.81243i 0.422813 + 0.906217i \(0.361043\pi\)
−0.422813 + 0.906217i \(0.638957\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 9.79796i 0.705273i −0.935760 0.352636i \(-0.885285\pi\)
0.935760 0.352636i \(-0.114715\pi\)
\(194\) 27.7128i 1.98966i
\(195\) 0 0
\(196\) −34.0000 −2.42857
\(197\) 14.1421 1.00759 0.503793 0.863825i \(-0.331938\pi\)
0.503793 + 0.863825i \(0.331938\pi\)
\(198\) 0 0
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) −2.82843 + 13.8564i −0.200000 + 0.979796i
\(201\) 0 0
\(202\) 4.89898i 0.344691i
\(203\) 50.9117 3.57330
\(204\) 0 0
\(205\) 0 0
\(206\) 20.7846i 1.44813i
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) −28.2843 −1.94257
\(213\) 0 0
\(214\) 16.0000 1.09374
\(215\) 0 0
\(216\) 0 0
\(217\) 48.9898i 3.32564i
\(218\) 0 0
\(219\) 0 0
\(220\) 12.0000 9.79796i 0.809040 0.660578i
\(221\) 0 0
\(222\) 0 0
\(223\) 14.6969i 0.984180i 0.870544 + 0.492090i \(0.163767\pi\)
−0.870544 + 0.492090i \(0.836233\pi\)
\(224\) 27.7128i 1.85164i
\(225\) 0 0
\(226\) 0 0
\(227\) −28.2843 −1.87729 −0.938647 0.344881i \(-0.887919\pi\)
−0.938647 + 0.344881i \(0.887919\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 29.3939i 1.92980i
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 20.7846i 1.35296i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −1.41421 −0.0909091
\(243\) 0 0
\(244\) 0 0
\(245\) 24.0416 + 29.4449i 1.53596 + 1.88116i
\(246\) 0 0
\(247\) 0 0
\(248\) 28.2843 1.79605
\(249\) 0 0
\(250\) 14.0000 7.34847i 0.885438 0.464758i
\(251\) 31.1769i 1.96787i −0.178529 0.983935i \(-0.557134\pi\)
0.178529 0.983935i \(-0.442866\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 6.92820i 0.434714i
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 4.89898i 0.302660i
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 20.0000 + 24.4949i 1.22859 + 1.50471i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 10.3923i 0.633630i −0.948487 0.316815i \(-0.897387\pi\)
0.948487 0.316815i \(-0.102613\pi\)
\(270\) 0 0
\(271\) −22.0000 −1.33640 −0.668202 0.743980i \(-0.732936\pi\)
−0.668202 + 0.743980i \(0.732936\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −16.9706 3.46410i −1.02336 0.208893i
\(276\) 0 0
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) −24.0000 + 19.5959i −1.43427 + 1.17108i
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) 25.4558 20.7846i 1.49482 1.22051i
\(291\) 0 0
\(292\) 19.5959i 1.14676i
\(293\) −14.1421 −0.826192 −0.413096 0.910687i \(-0.635553\pi\)
−0.413096 + 0.910687i \(0.635553\pi\)
\(294\) 0 0
\(295\) −18.0000 + 14.6969i −1.04800 + 0.855689i
\(296\) 0 0
\(297\) 0 0
\(298\) 34.2929i 1.98653i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 2.82843 0.162758
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 33.9411 1.93398
\(309\) 0 0
\(310\) −20.0000 24.4949i −1.13592 1.39122i
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 9.79796i 0.553813i −0.960897 0.276907i \(-0.910691\pi\)
0.960897 0.276907i \(-0.0893093\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −20.0000 −1.12509
\(317\) −31.1127 −1.74746 −0.873732 0.486408i \(-0.838307\pi\)
−0.873732 + 0.486408i \(0.838307\pi\)
\(318\) 0 0
\(319\) −36.0000 −2.01561
\(320\) −11.3137 13.8564i −0.632456 0.774597i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 11.3137 0.620920
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 29.3939i 1.60119i 0.599208 + 0.800593i \(0.295482\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −18.3848 −1.00000
\(339\) 0 0
\(340\) 0 0
\(341\) 34.6410i 1.87592i
\(342\) 0 0
\(343\) 48.9898i 2.64520i
\(344\) 0 0
\(345\) 0 0
\(346\) −28.0000 −1.50529
\(347\) −28.2843 −1.51838 −0.759190 0.650870i \(-0.774404\pi\)
−0.759190 + 0.650870i \(0.774404\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 33.9411 + 6.92820i 1.81423 + 0.370328i
\(351\) 0 0
\(352\) 19.5959i 1.04447i
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 34.2929i 1.81243i
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 16.9706 13.8564i 0.888280 0.725277i
\(366\) 0 0
\(367\) 4.89898i 0.255725i 0.991792 + 0.127862i \(0.0408116\pi\)
−0.991792 + 0.127862i \(0.959188\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 69.2820i 3.59694i
\(372\) 0 0
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) −24.0000 29.3939i −1.22315 1.49805i
\(386\) 13.8564i 0.705273i
\(387\) 0 0
\(388\) 39.1918i 1.98966i
\(389\) 24.2487i 1.22946i 0.788738 + 0.614729i \(0.210735\pi\)
−0.788738 + 0.614729i \(0.789265\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −48.0833 −2.42857
\(393\) 0 0
\(394\) 20.0000 1.00759
\(395\) 14.1421 + 17.3205i 0.711568 + 0.871489i
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) −19.7990 −0.992434
\(399\) 0 0
\(400\) −4.00000 + 19.5959i −0.200000 + 0.979796i
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 6.92820i 0.344691i
\(405\) 0 0
\(406\) 72.0000 3.57330
\(407\) 0 0
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 29.3939i 1.44813i
\(413\) −50.9117 −2.50520
\(414\) 0 0
\(415\) −8.00000 9.79796i −0.392705 0.480963i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 10.3923i 0.507697i 0.967244 + 0.253849i \(0.0816965\pi\)
−0.967244 + 0.253849i \(0.918303\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −40.0000 −1.94257
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 22.6274 1.09374
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 39.1918i 1.88344i −0.336399 0.941720i \(-0.609209\pi\)
0.336399 0.941720i \(-0.390791\pi\)
\(434\) 69.2820i 3.32564i
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 34.0000 1.62273 0.811366 0.584539i \(-0.198725\pi\)
0.811366 + 0.584539i \(0.198725\pi\)
\(440\) 16.9706 13.8564i 0.809040 0.660578i
\(441\) 0 0
\(442\) 0 0
\(443\) 28.2843 1.34383 0.671913 0.740630i \(-0.265473\pi\)
0.671913 + 0.740630i \(0.265473\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 20.7846i 0.984180i
\(447\) 0 0
\(448\) 39.1918i 1.85164i
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) −40.0000 −1.87729
\(455\) 0 0
\(456\) 0 0
\(457\) 19.5959i 0.916658i −0.888783 0.458329i \(-0.848448\pi\)
0.888783 0.458329i \(-0.151552\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 38.1051i 1.77473i −0.461065 0.887366i \(-0.652533\pi\)
0.461065 0.887366i \(-0.347467\pi\)
\(462\) 0 0
\(463\) 34.2929i 1.59372i −0.604161 0.796862i \(-0.706492\pi\)
0.604161 0.796862i \(-0.293508\pi\)
\(464\) 41.5692i 1.92980i
\(465\) 0 0
\(466\) 0 0
\(467\) −39.5980 −1.83238 −0.916188 0.400749i \(-0.868750\pi\)
−0.916188 + 0.400749i \(0.868750\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 29.3939i 1.35296i
\(473\) 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
\(481\) 0 0
\(482\) 14.1421 0.644157
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 33.9411 27.7128i 1.54119 1.25837i
\(486\) 0 0
\(487\) 44.0908i 1.99795i 0.0453143 + 0.998973i \(0.485571\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 34.0000 + 41.6413i 1.53596 + 1.88116i
\(491\) 38.1051i 1.71966i −0.510581 0.859830i \(-0.670569\pi\)
0.510581 0.859830i \(-0.329431\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 40.0000 1.79605
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 19.7990 10.3923i 0.885438 0.464758i
\(501\) 0 0
\(502\) 44.0908i 1.96787i
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) −6.00000 + 4.89898i −0.266996 + 0.218002i
\(506\) 0 0
\(507\) 0 0
\(508\) 9.79796i 0.434714i
\(509\) 45.0333i 1.99607i 0.0626839 + 0.998033i \(0.480034\pi\)
−0.0626839 + 0.998033i \(0.519966\pi\)
\(510\) 0 0
\(511\) 48.0000 2.12339
\(512\) 22.6274 1.00000
\(513\) 0 0
\(514\) 0 0
\(515\) −25.4558 + 20.7846i −1.12172 + 0.915879i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 6.92820i 0.302660i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 28.2843 + 34.6410i 1.22859 + 1.50471i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −16.0000 19.5959i −0.691740 0.847205i
\(536\) 0 0
\(537\) 0 0
\(538\) 14.6969i 0.633630i
\(539\) 58.8897i 2.53656i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) −31.1127 −1.33640
\(543\) 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
\(550\) −24.0000 4.89898i −1.02336 0.208893i
\(551\) 0 0
\(552\) 0 0
\(553\) 48.9898i 2.08326i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14.1421 0.599222 0.299611 0.954062i \(-0.403143\pi\)
0.299611 + 0.954062i \(0.403143\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −33.9411 + 27.7128i −1.43427 + 1.17108i
\(561\) 0 0
\(562\) 0 0
\(563\) 28.2843 1.19204 0.596020 0.802970i \(-0.296748\pi\)
0.596020 + 0.802970i \(0.296748\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 29.3939i 1.22368i 0.790980 + 0.611842i \(0.209571\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 24.0416 1.00000
\(579\) 0 0
\(580\) 36.0000 29.3939i 1.49482 1.22051i
\(581\) 27.7128i 1.14972i
\(582\) 0 0
\(583\) 48.9898i 2.02895i
\(584\) 27.7128i 1.14676i
\(585\) 0 0
\(586\) −20.0000 −0.826192
\(587\) 45.2548 1.86787 0.933933 0.357447i \(-0.116353\pi\)
0.933933 + 0.357447i \(0.116353\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −25.4558 + 20.7846i −1.04800 + 0.855689i
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 48.4974i 1.98653i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 4.00000 0.162758
\(605\) 1.41421 + 1.73205i 0.0574960 + 0.0704179i
\(606\) 0 0
\(607\) 44.0908i 1.78959i −0.446476 0.894795i \(-0.647321\pi\)
0.446476 0.894795i \(-0.352679\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 48.0000 1.93398
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) −28.2843 34.6410i −1.13592 1.39122i
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −23.0000 9.79796i −0.920000 0.391918i
\(626\) 13.8564i 0.553813i
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −50.0000 −1.99047 −0.995234 0.0975126i \(-0.968911\pi\)
−0.995234 + 0.0975126i \(0.968911\pi\)
\(632\) −28.2843 −1.12509
\(633\) 0 0
\(634\) −44.0000 −1.74746
\(635\) 8.48528 6.92820i 0.336728 0.274937i
\(636\) 0 0
\(637\) 0 0
\(638\) −50.9117 −2.01561
\(639\) 0 0
\(640\) −16.0000 19.5959i −0.632456 0.774597i
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 48.0833 1.88164 0.940822 0.338902i \(-0.110055\pi\)
0.940822 + 0.338902i \(0.110055\pi\)
\(654\) 0 0
\(655\) −6.00000 + 4.89898i −0.234439 + 0.191419i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 24.2487i 0.944596i −0.881439 0.472298i \(-0.843425\pi\)
0.881439 0.472298i \(-0.156575\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 16.0000 0.620920
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 39.1918i 1.51073i 0.655302 + 0.755367i \(0.272541\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 41.5692i 1.60119i
\(675\) 0 0
\(676\) −26.0000 −1.00000
\(677\) 2.82843 0.108705 0.0543526 0.998522i \(-0.482690\pi\)
0.0543526 + 0.998522i \(0.482690\pi\)
\(678\) 0 0
\(679\) 96.0000 3.68414
\(680\) 0 0
\(681\) 0 0
\(682\) 48.9898i 1.87592i
\(683\) 5.65685 0.216454 0.108227 0.994126i \(-0.465483\pi\)
0.108227 + 0.994126i \(0.465483\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 69.2820i 2.64520i
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) −39.5980 −1.50529
\(693\) 0 0
\(694\) −40.0000 −1.51838
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 48.0000 + 9.79796i 1.81423 + 0.370328i
\(701\) 38.1051i 1.43921i 0.694383 + 0.719605i \(0.255677\pi\)
−0.694383 + 0.719605i \(0.744323\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 27.7128i 1.04447i
\(705\) 0 0
\(706\) 0 0
\(707\) −16.9706 −0.638244
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 48.4974i 1.81243i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −72.0000 −2.68142
\(722\) 26.8701 1.00000
\(723\) 0 0
\(724\) 0 0
\(725\) −50.9117 10.3923i −1.89081 0.385961i
\(726\) 0 0
\(727\) 53.8888i 1.99862i −0.0370879 0.999312i \(-0.511808\pi\)
0.0370879 0.999312i \(-0.488192\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 24.0000 19.5959i 0.888280 0.725277i
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 6.92820i 0.255725i
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 97.9796i 3.59694i
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −42.0000 + 34.2929i −1.53876 + 1.25639i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 55.4256i 2.02521i
\(750\) 0 0
\(751\) 10.0000 0.364905 0.182453 0.983215i \(-0.441596\pi\)
0.182453 + 0.983215i \(0.441596\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −2.82843 3.46410i −0.102937 0.126072i
\(756\) 0 0
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −26.0000 −0.937584 −0.468792 0.883309i \(-0.655311\pi\)
−0.468792 + 0.883309i \(0.655311\pi\)
\(770\) −33.9411 41.5692i −1.22315 1.49805i
\(771\) 0 0
\(772\) 19.5959i 0.705273i
\(773\) −19.7990 −0.712120 −0.356060 0.934463i \(-0.615880\pi\)
−0.356060 + 0.934463i \(0.615880\pi\)
\(774\) 0 0
\(775\) −10.0000 + 48.9898i −0.359211 + 1.75977i
\(776\) 55.4256i 1.98966i
\(777\) 0 0
\(778\) 34.2929i 1.22946i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −68.0000 −2.42857
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 28.2843 1.00759
\(789\) 0 0
\(790\) 20.0000 + 24.4949i 0.711568 + 0.871489i
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −28.0000 −0.992434
\(797\) 53.7401 1.90357 0.951786 0.306762i \(-0.0992455\pi\)
0.951786 + 0.306762i \(0.0992455\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −5.65685 + 27.7128i −0.200000 + 0.979796i
\(801\) 0 0
\(802\) 0 0
\(803\) −33.9411 −1.19776
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 9.79796i 0.344691i
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 101.823 3.57330
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −14.1421 −0.494468
\(819\) 0 0
\(820\) 0 0
\(821\) 31.1769i 1.08808i −0.839059 0.544041i \(-0.816894\pi\)
0.839059 0.544041i \(-0.183106\pi\)
\(822\) 0 0
\(823\) 34.2929i 1.19537i 0.801730 + 0.597687i \(0.203913\pi\)
−0.801730 + 0.597687i \(0.796087\pi\)
\(824\) 41.5692i 1.44813i
\(825\) 0 0
\(826\) −72.0000 −2.50520
\(827\) 56.5685 1.96708 0.983540 0.180688i \(-0.0578324\pi\)
0.983540 + 0.180688i \(0.0578324\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) −11.3137 13.8564i −0.392705 0.480963i
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 14.6969i 0.507697i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −79.0000 −2.72414
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 18.3848 + 22.5167i 0.632456 + 0.774597i
\(846\) 0 0
\(847\) 4.89898i 0.168331i
\(848\) −56.5685 −1.94257
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 32.0000 1.09374
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 28.0000 + 34.2929i 0.952029 + 1.16599i
\(866\) 55.4256i 1.88344i
\(867\) 0 0
\(868\) 97.9796i 3.32564i
\(869\) 34.6410i 1.17512i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −25.4558 48.4974i −0.860565 1.63951i
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 48.0833 1.62273
\(879\) 0 0
\(880\) 24.0000 19.5959i 0.809040 0.660578i
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 40.0000 1.34383
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 24.0000 0.804934
\(890\) 0 0
\(891\) 0 0
\(892\) 29.3939i 0.984180i
\(893\) 0 0
\(894\) 0 0
\(895\) 42.0000 34.2929i 1.40391 1.14628i
\(896\) 55.4256i 1.85164i
\(897\) 0 0
\(898\) 0 0
\(899\) 103.923i 3.46603i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) −56.5685 −1.87729
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 19.5959i 0.648530i
\(914\) 27.7128i 0.916658i
\(915\) 0 0
\(916\) 0 0
\(917\) −16.9706 −0.560417
\(918\) 0 0
\(919\) 50.0000 1.64935 0.824674 0.565608i \(-0.191359\pi\)
0.824674 + 0.565608i \(0.191359\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 53.8888i 1.77473i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 48.4974i 1.59372i
\(927\) 0 0
\(928\) 58.7878i 1.92980i
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) −56.0000 −1.83238
\(935\) 0 0
\(936\) 0 0
\(937\) 19.5959i 0.640171i 0.947389 + 0.320085i \(0.103712\pi\)
−0.947389 + 0.320085i \(0.896288\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 38.1051i 1.24219i 0.783735 + 0.621096i \(0.213312\pi\)
−0.783735 + 0.621096i \(0.786688\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 41.5692i 1.35296i
\(945\) 0 0
\(946\) 0 0
\(947\) 56.5685 1.83823 0.919115 0.393989i \(-0.128905\pi\)
0.919115 + 0.393989i \(0.128905\pi\)
\(948\) 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
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) 0 0
\(964\) 20.0000 0.644157
\(965\) −16.9706 + 13.8564i −0.546302 + 0.446054i
\(966\) 0 0
\(967\) 4.89898i 0.157541i −0.996893 0.0787703i \(-0.974901\pi\)
0.996893 0.0787703i \(-0.0250994\pi\)
\(968\) −2.82843 −0.0909091
\(969\) 0 0
\(970\) 48.0000 39.1918i 1.54119 1.25837i
\(971\) 3.46410i 0.111168i −0.998454 0.0555842i \(-0.982298\pi\)
0.998454 0.0555842i \(-0.0177021\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 62.3538i 1.99795i
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 48.0833 + 58.8897i 1.53596 + 1.88116i
\(981\) 0 0
\(982\) 53.8888i 1.71966i
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) −20.0000 24.4949i −0.637253 0.780472i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −58.0000 −1.84243 −0.921215 0.389053i \(-0.872802\pi\)
−0.921215 + 0.389053i \(0.872802\pi\)
\(992\) 56.5685 1.79605
\(993\) 0 0
\(994\) 0 0
\(995\) 19.7990 + 24.2487i 0.627670 + 0.768736i
\(996\) 0 0
\(997\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) 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 360.2.d.d.109.3 yes 4
3.2 odd 2 inner 360.2.d.d.109.2 yes 4
4.3 odd 2 1440.2.d.b.1009.1 4
5.2 odd 4 1800.2.k.k.901.4 4
5.3 odd 4 1800.2.k.k.901.1 4
5.4 even 2 inner 360.2.d.d.109.1 4
8.3 odd 2 1440.2.d.b.1009.4 4
8.5 even 2 inner 360.2.d.d.109.2 yes 4
12.11 even 2 1440.2.d.b.1009.4 4
15.2 even 4 1800.2.k.k.901.2 4
15.8 even 4 1800.2.k.k.901.3 4
15.14 odd 2 inner 360.2.d.d.109.4 yes 4
20.3 even 4 7200.2.k.k.3601.3 4
20.7 even 4 7200.2.k.k.3601.1 4
20.19 odd 2 1440.2.d.b.1009.3 4
24.5 odd 2 CM 360.2.d.d.109.3 yes 4
24.11 even 2 1440.2.d.b.1009.1 4
40.3 even 4 7200.2.k.k.3601.4 4
40.13 odd 4 1800.2.k.k.901.3 4
40.19 odd 2 1440.2.d.b.1009.2 4
40.27 even 4 7200.2.k.k.3601.2 4
40.29 even 2 inner 360.2.d.d.109.4 yes 4
40.37 odd 4 1800.2.k.k.901.2 4
60.23 odd 4 7200.2.k.k.3601.4 4
60.47 odd 4 7200.2.k.k.3601.2 4
60.59 even 2 1440.2.d.b.1009.2 4
120.29 odd 2 inner 360.2.d.d.109.1 4
120.53 even 4 1800.2.k.k.901.1 4
120.59 even 2 1440.2.d.b.1009.3 4
120.77 even 4 1800.2.k.k.901.4 4
120.83 odd 4 7200.2.k.k.3601.3 4
120.107 odd 4 7200.2.k.k.3601.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.d.d.109.1 4 5.4 even 2 inner
360.2.d.d.109.1 4 120.29 odd 2 inner
360.2.d.d.109.2 yes 4 3.2 odd 2 inner
360.2.d.d.109.2 yes 4 8.5 even 2 inner
360.2.d.d.109.3 yes 4 1.1 even 1 trivial
360.2.d.d.109.3 yes 4 24.5 odd 2 CM
360.2.d.d.109.4 yes 4 15.14 odd 2 inner
360.2.d.d.109.4 yes 4 40.29 even 2 inner
1440.2.d.b.1009.1 4 4.3 odd 2
1440.2.d.b.1009.1 4 24.11 even 2
1440.2.d.b.1009.2 4 40.19 odd 2
1440.2.d.b.1009.2 4 60.59 even 2
1440.2.d.b.1009.3 4 20.19 odd 2
1440.2.d.b.1009.3 4 120.59 even 2
1440.2.d.b.1009.4 4 8.3 odd 2
1440.2.d.b.1009.4 4 12.11 even 2
1800.2.k.k.901.1 4 5.3 odd 4
1800.2.k.k.901.1 4 120.53 even 4
1800.2.k.k.901.2 4 15.2 even 4
1800.2.k.k.901.2 4 40.37 odd 4
1800.2.k.k.901.3 4 15.8 even 4
1800.2.k.k.901.3 4 40.13 odd 4
1800.2.k.k.901.4 4 5.2 odd 4
1800.2.k.k.901.4 4 120.77 even 4
7200.2.k.k.3601.1 4 20.7 even 4
7200.2.k.k.3601.1 4 120.107 odd 4
7200.2.k.k.3601.2 4 40.27 even 4
7200.2.k.k.3601.2 4 60.47 odd 4
7200.2.k.k.3601.3 4 20.3 even 4
7200.2.k.k.3601.3 4 120.83 odd 4
7200.2.k.k.3601.4 4 40.3 even 4
7200.2.k.k.3601.4 4 60.23 odd 4