Properties

Label 630.2.d.b.629.4
Level $630$
Weight $2$
Character 630.629
Analytic conductor $5.031$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [630,2,Mod(629,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.629");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 630.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: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 629.4
Root \(-0.874032i\) of defining polynomial
Character \(\chi\) \(=\) 630.629
Dual form 630.2.d.b.629.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.23607 q^{5} +(-2.23607 + 1.41421i) q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.23607 q^{5} +(-2.23607 + 1.41421i) q^{7} -1.00000 q^{8} -2.23607 q^{10} -5.65685i q^{11} +4.47214 q^{13} +(2.23607 - 1.41421i) q^{14} +1.00000 q^{16} -3.16228i q^{17} +3.16228i q^{19} +2.23607 q^{20} +5.65685i q^{22} +4.00000 q^{23} +5.00000 q^{25} -4.47214 q^{26} +(-2.23607 + 1.41421i) q^{28} -2.82843i q^{29} +6.32456i q^{31} -1.00000 q^{32} +3.16228i q^{34} +(-5.00000 + 3.16228i) q^{35} -9.89949i q^{37} -3.16228i q^{38} -2.23607 q^{40} +4.47214 q^{41} -1.41421i q^{43} -5.65685i q^{44} -4.00000 q^{46} +9.48683i q^{47} +(3.00000 - 6.32456i) q^{49} -5.00000 q^{50} +4.47214 q^{52} +4.00000 q^{53} -12.6491i q^{55} +(2.23607 - 1.41421i) q^{56} +2.82843i q^{58} +4.47214 q^{59} +9.48683i q^{61} -6.32456i q^{62} +1.00000 q^{64} +10.0000 q^{65} -7.07107i q^{67} -3.16228i q^{68} +(5.00000 - 3.16228i) q^{70} -1.41421i q^{71} +13.4164 q^{73} +9.89949i q^{74} +3.16228i q^{76} +(8.00000 + 12.6491i) q^{77} +6.00000 q^{79} +2.23607 q^{80} -4.47214 q^{82} -12.6491i q^{83} -7.07107i q^{85} +1.41421i q^{86} +5.65685i q^{88} -4.47214 q^{89} +(-10.0000 + 6.32456i) q^{91} +4.00000 q^{92} -9.48683i q^{94} +7.07107i q^{95} +(-3.00000 + 6.32456i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 4 q^{16} + 16 q^{23} + 20 q^{25} - 4 q^{32} - 20 q^{35} - 16 q^{46} + 12 q^{49} - 20 q^{50} + 16 q^{53} + 4 q^{64} + 40 q^{65} + 20 q^{70} + 32 q^{77} + 24 q^{79} - 40 q^{91} + 16 q^{92} - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 2.23607 1.00000
\(6\) 0 0
\(7\) −2.23607 + 1.41421i −0.845154 + 0.534522i
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −2.23607 −0.707107
\(11\) 5.65685i 1.70561i −0.522233 0.852803i \(-0.674901\pi\)
0.522233 0.852803i \(-0.325099\pi\)
\(12\) 0 0
\(13\) 4.47214 1.24035 0.620174 0.784465i \(-0.287062\pi\)
0.620174 + 0.784465i \(0.287062\pi\)
\(14\) 2.23607 1.41421i 0.597614 0.377964i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.16228i 0.766965i −0.923548 0.383482i \(-0.874725\pi\)
0.923548 0.383482i \(-0.125275\pi\)
\(18\) 0 0
\(19\) 3.16228i 0.725476i 0.931891 + 0.362738i \(0.118158\pi\)
−0.931891 + 0.362738i \(0.881842\pi\)
\(20\) 2.23607 0.500000
\(21\) 0 0
\(22\) 5.65685i 1.20605i
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) −4.47214 −0.877058
\(27\) 0 0
\(28\) −2.23607 + 1.41421i −0.422577 + 0.267261i
\(29\) 2.82843i 0.525226i −0.964901 0.262613i \(-0.915416\pi\)
0.964901 0.262613i \(-0.0845842\pi\)
\(30\) 0 0
\(31\) 6.32456i 1.13592i 0.823055 + 0.567962i \(0.192268\pi\)
−0.823055 + 0.567962i \(0.807732\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 3.16228i 0.542326i
\(35\) −5.00000 + 3.16228i −0.845154 + 0.534522i
\(36\) 0 0
\(37\) 9.89949i 1.62747i −0.581238 0.813733i \(-0.697432\pi\)
0.581238 0.813733i \(-0.302568\pi\)
\(38\) 3.16228i 0.512989i
\(39\) 0 0
\(40\) −2.23607 −0.353553
\(41\) 4.47214 0.698430 0.349215 0.937043i \(-0.386448\pi\)
0.349215 + 0.937043i \(0.386448\pi\)
\(42\) 0 0
\(43\) 1.41421i 0.215666i −0.994169 0.107833i \(-0.965609\pi\)
0.994169 0.107833i \(-0.0343911\pi\)
\(44\) 5.65685i 0.852803i
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 9.48683i 1.38380i 0.721995 + 0.691898i \(0.243225\pi\)
−0.721995 + 0.691898i \(0.756775\pi\)
\(48\) 0 0
\(49\) 3.00000 6.32456i 0.428571 0.903508i
\(50\) −5.00000 −0.707107
\(51\) 0 0
\(52\) 4.47214 0.620174
\(53\) 4.00000 0.549442 0.274721 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 12.6491i 1.70561i
\(56\) 2.23607 1.41421i 0.298807 0.188982i
\(57\) 0 0
\(58\) 2.82843i 0.371391i
\(59\) 4.47214 0.582223 0.291111 0.956689i \(-0.405975\pi\)
0.291111 + 0.956689i \(0.405975\pi\)
\(60\) 0 0
\(61\) 9.48683i 1.21466i 0.794448 + 0.607332i \(0.207760\pi\)
−0.794448 + 0.607332i \(0.792240\pi\)
\(62\) 6.32456i 0.803219i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 10.0000 1.24035
\(66\) 0 0
\(67\) 7.07107i 0.863868i −0.901905 0.431934i \(-0.857831\pi\)
0.901905 0.431934i \(-0.142169\pi\)
\(68\) 3.16228i 0.383482i
\(69\) 0 0
\(70\) 5.00000 3.16228i 0.597614 0.377964i
\(71\) 1.41421i 0.167836i −0.996473 0.0839181i \(-0.973257\pi\)
0.996473 0.0839181i \(-0.0267434\pi\)
\(72\) 0 0
\(73\) 13.4164 1.57027 0.785136 0.619324i \(-0.212593\pi\)
0.785136 + 0.619324i \(0.212593\pi\)
\(74\) 9.89949i 1.15079i
\(75\) 0 0
\(76\) 3.16228i 0.362738i
\(77\) 8.00000 + 12.6491i 0.911685 + 1.44150i
\(78\) 0 0
\(79\) 6.00000 0.675053 0.337526 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 2.23607 0.250000
\(81\) 0 0
\(82\) −4.47214 −0.493865
\(83\) 12.6491i 1.38842i −0.719772 0.694210i \(-0.755754\pi\)
0.719772 0.694210i \(-0.244246\pi\)
\(84\) 0 0
\(85\) 7.07107i 0.766965i
\(86\) 1.41421i 0.152499i
\(87\) 0 0
\(88\) 5.65685i 0.603023i
\(89\) −4.47214 −0.474045 −0.237023 0.971504i \(-0.576172\pi\)
−0.237023 + 0.971504i \(0.576172\pi\)
\(90\) 0 0
\(91\) −10.0000 + 6.32456i −1.04828 + 0.662994i
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) 9.48683i 0.978492i
\(95\) 7.07107i 0.725476i
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) −3.00000 + 6.32456i −0.303046 + 0.638877i
\(99\) 0 0
\(100\) 5.00000 0.500000
\(101\) −17.8885 −1.77998 −0.889988 0.455983i \(-0.849288\pi\)
−0.889988 + 0.455983i \(0.849288\pi\)
\(102\) 0 0
\(103\) −8.94427 −0.881305 −0.440653 0.897678i \(-0.645253\pi\)
−0.440653 + 0.897678i \(0.645253\pi\)
\(104\) −4.47214 −0.438529
\(105\) 0 0
\(106\) −4.00000 −0.388514
\(107\) −18.0000 −1.74013 −0.870063 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 12.6491i 1.20605i
\(111\) 0 0
\(112\) −2.23607 + 1.41421i −0.211289 + 0.133631i
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 8.94427 0.834058
\(116\) 2.82843i 0.262613i
\(117\) 0 0
\(118\) −4.47214 −0.411693
\(119\) 4.47214 + 7.07107i 0.409960 + 0.648204i
\(120\) 0 0
\(121\) −21.0000 −1.90909
\(122\) 9.48683i 0.858898i
\(123\) 0 0
\(124\) 6.32456i 0.567962i
\(125\) 11.1803 1.00000
\(126\) 0 0
\(127\) 14.1421i 1.25491i 0.778652 + 0.627456i \(0.215904\pi\)
−0.778652 + 0.627456i \(0.784096\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −10.0000 −0.877058
\(131\) −17.8885 −1.56293 −0.781465 0.623949i \(-0.785527\pi\)
−0.781465 + 0.623949i \(0.785527\pi\)
\(132\) 0 0
\(133\) −4.47214 7.07107i −0.387783 0.613139i
\(134\) 7.07107i 0.610847i
\(135\) 0 0
\(136\) 3.16228i 0.271163i
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) 15.8114i 1.34110i −0.741862 0.670552i \(-0.766057\pi\)
0.741862 0.670552i \(-0.233943\pi\)
\(140\) −5.00000 + 3.16228i −0.422577 + 0.267261i
\(141\) 0 0
\(142\) 1.41421i 0.118678i
\(143\) 25.2982i 2.11554i
\(144\) 0 0
\(145\) 6.32456i 0.525226i
\(146\) −13.4164 −1.11035
\(147\) 0 0
\(148\) 9.89949i 0.813733i
\(149\) 16.9706i 1.39028i 0.718873 + 0.695141i \(0.244658\pi\)
−0.718873 + 0.695141i \(0.755342\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 3.16228i 0.256495i
\(153\) 0 0
\(154\) −8.00000 12.6491i −0.644658 1.01929i
\(155\) 14.1421i 1.13592i
\(156\) 0 0
\(157\) 4.47214 0.356915 0.178458 0.983948i \(-0.442889\pi\)
0.178458 + 0.983948i \(0.442889\pi\)
\(158\) −6.00000 −0.477334
\(159\) 0 0
\(160\) −2.23607 −0.176777
\(161\) −8.94427 + 5.65685i −0.704907 + 0.445823i
\(162\) 0 0
\(163\) 21.2132i 1.66155i 0.556611 + 0.830773i \(0.312101\pi\)
−0.556611 + 0.830773i \(0.687899\pi\)
\(164\) 4.47214 0.349215
\(165\) 0 0
\(166\) 12.6491i 0.981761i
\(167\) 9.48683i 0.734113i 0.930199 + 0.367057i \(0.119634\pi\)
−0.930199 + 0.367057i \(0.880366\pi\)
\(168\) 0 0
\(169\) 7.00000 0.538462
\(170\) 7.07107i 0.542326i
\(171\) 0 0
\(172\) 1.41421i 0.107833i
\(173\) 25.2982i 1.92339i 0.274125 + 0.961694i \(0.411612\pi\)
−0.274125 + 0.961694i \(0.588388\pi\)
\(174\) 0 0
\(175\) −11.1803 + 7.07107i −0.845154 + 0.534522i
\(176\) 5.65685i 0.426401i
\(177\) 0 0
\(178\) 4.47214 0.335201
\(179\) 2.82843i 0.211407i −0.994398 0.105703i \(-0.966291\pi\)
0.994398 0.105703i \(-0.0337094\pi\)
\(180\) 0 0
\(181\) 3.16228i 0.235050i −0.993070 0.117525i \(-0.962504\pi\)
0.993070 0.117525i \(-0.0374961\pi\)
\(182\) 10.0000 6.32456i 0.741249 0.468807i
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) 22.1359i 1.62747i
\(186\) 0 0
\(187\) −17.8885 −1.30814
\(188\) 9.48683i 0.691898i
\(189\) 0 0
\(190\) 7.07107i 0.512989i
\(191\) 1.41421i 0.102329i 0.998690 + 0.0511645i \(0.0162933\pi\)
−0.998690 + 0.0511645i \(0.983707\pi\)
\(192\) 0 0
\(193\) 8.48528i 0.610784i 0.952227 + 0.305392i \(0.0987875\pi\)
−0.952227 + 0.305392i \(0.901213\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 3.00000 6.32456i 0.214286 0.451754i
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −5.00000 −0.353553
\(201\) 0 0
\(202\) 17.8885 1.25863
\(203\) 4.00000 + 6.32456i 0.280745 + 0.443897i
\(204\) 0 0
\(205\) 10.0000 0.698430
\(206\) 8.94427 0.623177
\(207\) 0 0
\(208\) 4.47214 0.310087
\(209\) 17.8885 1.23738
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 4.00000 0.274721
\(213\) 0 0
\(214\) 18.0000 1.23045
\(215\) 3.16228i 0.215666i
\(216\) 0 0
\(217\) −8.94427 14.1421i −0.607177 0.960031i
\(218\) 10.0000 0.677285
\(219\) 0 0
\(220\) 12.6491i 0.852803i
\(221\) 14.1421i 0.951303i
\(222\) 0 0
\(223\) 8.94427 0.598953 0.299476 0.954104i \(-0.403188\pi\)
0.299476 + 0.954104i \(0.403188\pi\)
\(224\) 2.23607 1.41421i 0.149404 0.0944911i
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 18.9737i 1.25933i −0.776868 0.629663i \(-0.783193\pi\)
0.776868 0.629663i \(-0.216807\pi\)
\(228\) 0 0
\(229\) 15.8114i 1.04485i −0.852686 0.522423i \(-0.825028\pi\)
0.852686 0.522423i \(-0.174972\pi\)
\(230\) −8.94427 −0.589768
\(231\) 0 0
\(232\) 2.82843i 0.185695i
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) 0 0
\(235\) 21.2132i 1.38380i
\(236\) 4.47214 0.291111
\(237\) 0 0
\(238\) −4.47214 7.07107i −0.289886 0.458349i
\(239\) 4.24264i 0.274434i −0.990541 0.137217i \(-0.956184\pi\)
0.990541 0.137217i \(-0.0438157\pi\)
\(240\) 0 0
\(241\) 25.2982i 1.62960i 0.579741 + 0.814801i \(0.303154\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 21.0000 1.34993
\(243\) 0 0
\(244\) 9.48683i 0.607332i
\(245\) 6.70820 14.1421i 0.428571 0.903508i
\(246\) 0 0
\(247\) 14.1421i 0.899843i
\(248\) 6.32456i 0.401610i
\(249\) 0 0
\(250\) −11.1803 −0.707107
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 22.6274i 1.42257i
\(254\) 14.1421i 0.887357i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 15.8114i 0.986287i 0.869948 + 0.493144i \(0.164152\pi\)
−0.869948 + 0.493144i \(0.835848\pi\)
\(258\) 0 0
\(259\) 14.0000 + 22.1359i 0.869918 + 1.37546i
\(260\) 10.0000 0.620174
\(261\) 0 0
\(262\) 17.8885 1.10516
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) 8.94427 0.549442
\(266\) 4.47214 + 7.07107i 0.274204 + 0.433555i
\(267\) 0 0
\(268\) 7.07107i 0.431934i
\(269\) −17.8885 −1.09068 −0.545342 0.838214i \(-0.683600\pi\)
−0.545342 + 0.838214i \(0.683600\pi\)
\(270\) 0 0
\(271\) 6.32456i 0.384189i 0.981376 + 0.192095i \(0.0615281\pi\)
−0.981376 + 0.192095i \(0.938472\pi\)
\(272\) 3.16228i 0.191741i
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) 28.2843i 1.70561i
\(276\) 0 0
\(277\) 21.2132i 1.27458i −0.770625 0.637289i \(-0.780056\pi\)
0.770625 0.637289i \(-0.219944\pi\)
\(278\) 15.8114i 0.948304i
\(279\) 0 0
\(280\) 5.00000 3.16228i 0.298807 0.188982i
\(281\) 12.7279i 0.759284i −0.925133 0.379642i \(-0.876047\pi\)
0.925133 0.379642i \(-0.123953\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 1.41421i 0.0839181i
\(285\) 0 0
\(286\) 25.2982i 1.49592i
\(287\) −10.0000 + 6.32456i −0.590281 + 0.373327i
\(288\) 0 0
\(289\) 7.00000 0.411765
\(290\) 6.32456i 0.371391i
\(291\) 0 0
\(292\) 13.4164 0.785136
\(293\) 6.32456i 0.369484i 0.982787 + 0.184742i \(0.0591450\pi\)
−0.982787 + 0.184742i \(0.940855\pi\)
\(294\) 0 0
\(295\) 10.0000 0.582223
\(296\) 9.89949i 0.575396i
\(297\) 0 0
\(298\) 16.9706i 0.983078i
\(299\) 17.8885 1.03452
\(300\) 0 0
\(301\) 2.00000 + 3.16228i 0.115278 + 0.182271i
\(302\) −10.0000 −0.575435
\(303\) 0 0
\(304\) 3.16228i 0.181369i
\(305\) 21.2132i 1.21466i
\(306\) 0 0
\(307\) −26.8328 −1.53143 −0.765715 0.643180i \(-0.777615\pi\)
−0.765715 + 0.643180i \(0.777615\pi\)
\(308\) 8.00000 + 12.6491i 0.455842 + 0.720750i
\(309\) 0 0
\(310\) 14.1421i 0.803219i
\(311\) 8.94427 0.507183 0.253592 0.967311i \(-0.418388\pi\)
0.253592 + 0.967311i \(0.418388\pi\)
\(312\) 0 0
\(313\) −13.4164 −0.758340 −0.379170 0.925327i \(-0.623790\pi\)
−0.379170 + 0.925327i \(0.623790\pi\)
\(314\) −4.47214 −0.252377
\(315\) 0 0
\(316\) 6.00000 0.337526
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) −16.0000 −0.895828
\(320\) 2.23607 0.125000
\(321\) 0 0
\(322\) 8.94427 5.65685i 0.498445 0.315244i
\(323\) 10.0000 0.556415
\(324\) 0 0
\(325\) 22.3607 1.24035
\(326\) 21.2132i 1.17489i
\(327\) 0 0
\(328\) −4.47214 −0.246932
\(329\) −13.4164 21.2132i −0.739671 1.16952i
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 12.6491i 0.694210i
\(333\) 0 0
\(334\) 9.48683i 0.519096i
\(335\) 15.8114i 0.863868i
\(336\) 0 0
\(337\) 25.4558i 1.38667i 0.720616 + 0.693334i \(0.243859\pi\)
−0.720616 + 0.693334i \(0.756141\pi\)
\(338\) −7.00000 −0.380750
\(339\) 0 0
\(340\) 7.07107i 0.383482i
\(341\) 35.7771 1.93744
\(342\) 0 0
\(343\) 2.23607 + 18.3848i 0.120736 + 0.992685i
\(344\) 1.41421i 0.0762493i
\(345\) 0 0
\(346\) 25.2982i 1.36004i
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) 15.8114i 0.846364i −0.906045 0.423182i \(-0.860913\pi\)
0.906045 0.423182i \(-0.139087\pi\)
\(350\) 11.1803 7.07107i 0.597614 0.377964i
\(351\) 0 0
\(352\) 5.65685i 0.301511i
\(353\) 3.16228i 0.168311i −0.996453 0.0841555i \(-0.973181\pi\)
0.996453 0.0841555i \(-0.0268193\pi\)
\(354\) 0 0
\(355\) 3.16228i 0.167836i
\(356\) −4.47214 −0.237023
\(357\) 0 0
\(358\) 2.82843i 0.149487i
\(359\) 18.3848i 0.970311i −0.874428 0.485156i \(-0.838763\pi\)
0.874428 0.485156i \(-0.161237\pi\)
\(360\) 0 0
\(361\) 9.00000 0.473684
\(362\) 3.16228i 0.166206i
\(363\) 0 0
\(364\) −10.0000 + 6.32456i −0.524142 + 0.331497i
\(365\) 30.0000 1.57027
\(366\) 0 0
\(367\) 13.4164 0.700331 0.350165 0.936688i \(-0.386125\pi\)
0.350165 + 0.936688i \(0.386125\pi\)
\(368\) 4.00000 0.208514
\(369\) 0 0
\(370\) 22.1359i 1.15079i
\(371\) −8.94427 + 5.65685i −0.464363 + 0.293689i
\(372\) 0 0
\(373\) 21.2132i 1.09838i −0.835698 0.549189i \(-0.814937\pi\)
0.835698 0.549189i \(-0.185063\pi\)
\(374\) 17.8885 0.924995
\(375\) 0 0
\(376\) 9.48683i 0.489246i
\(377\) 12.6491i 0.651462i
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 7.07107i 0.362738i
\(381\) 0 0
\(382\) 1.41421i 0.0723575i
\(383\) 9.48683i 0.484755i −0.970182 0.242377i \(-0.922073\pi\)
0.970182 0.242377i \(-0.0779272\pi\)
\(384\) 0 0
\(385\) 17.8885 + 28.2843i 0.911685 + 1.44150i
\(386\) 8.48528i 0.431889i
\(387\) 0 0
\(388\) 0 0
\(389\) 16.9706i 0.860442i 0.902724 + 0.430221i \(0.141564\pi\)
−0.902724 + 0.430221i \(0.858436\pi\)
\(390\) 0 0
\(391\) 12.6491i 0.639693i
\(392\) −3.00000 + 6.32456i −0.151523 + 0.319438i
\(393\) 0 0
\(394\) 12.0000 0.604551
\(395\) 13.4164 0.675053
\(396\) 0 0
\(397\) −4.47214 −0.224450 −0.112225 0.993683i \(-0.535798\pi\)
−0.112225 + 0.993683i \(0.535798\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 5.00000 0.250000
\(401\) 1.41421i 0.0706225i 0.999376 + 0.0353112i \(0.0112422\pi\)
−0.999376 + 0.0353112i \(0.988758\pi\)
\(402\) 0 0
\(403\) 28.2843i 1.40894i
\(404\) −17.8885 −0.889988
\(405\) 0 0
\(406\) −4.00000 6.32456i −0.198517 0.313882i
\(407\) −56.0000 −2.77582
\(408\) 0 0
\(409\) 37.9473i 1.87637i 0.346128 + 0.938187i \(0.387496\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) −10.0000 −0.493865
\(411\) 0 0
\(412\) −8.94427 −0.440653
\(413\) −10.0000 + 6.32456i −0.492068 + 0.311211i
\(414\) 0 0
\(415\) 28.2843i 1.38842i
\(416\) −4.47214 −0.219265
\(417\) 0 0
\(418\) −17.8885 −0.874957
\(419\) 13.4164 0.655434 0.327717 0.944776i \(-0.393721\pi\)
0.327717 + 0.944776i \(0.393721\pi\)
\(420\) 0 0
\(421\) 38.0000 1.85201 0.926003 0.377515i \(-0.123221\pi\)
0.926003 + 0.377515i \(0.123221\pi\)
\(422\) 20.0000 0.973585
\(423\) 0 0
\(424\) −4.00000 −0.194257
\(425\) 15.8114i 0.766965i
\(426\) 0 0
\(427\) −13.4164 21.2132i −0.649265 1.02658i
\(428\) −18.0000 −0.870063
\(429\) 0 0
\(430\) 3.16228i 0.152499i
\(431\) 15.5563i 0.749323i −0.927162 0.374661i \(-0.877759\pi\)
0.927162 0.374661i \(-0.122241\pi\)
\(432\) 0 0
\(433\) −26.8328 −1.28950 −0.644751 0.764392i \(-0.723039\pi\)
−0.644751 + 0.764392i \(0.723039\pi\)
\(434\) 8.94427 + 14.1421i 0.429339 + 0.678844i
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 12.6491i 0.605089i
\(438\) 0 0
\(439\) 6.32456i 0.301855i 0.988545 + 0.150927i \(0.0482259\pi\)
−0.988545 + 0.150927i \(0.951774\pi\)
\(440\) 12.6491i 0.603023i
\(441\) 0 0
\(442\) 14.1421i 0.672673i
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) −10.0000 −0.474045
\(446\) −8.94427 −0.423524
\(447\) 0 0
\(448\) −2.23607 + 1.41421i −0.105644 + 0.0668153i
\(449\) 9.89949i 0.467186i −0.972334 0.233593i \(-0.924952\pi\)
0.972334 0.233593i \(-0.0750483\pi\)
\(450\) 0 0
\(451\) 25.2982i 1.19125i
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) 18.9737i 0.890478i
\(455\) −22.3607 + 14.1421i −1.04828 + 0.662994i
\(456\) 0 0
\(457\) 16.9706i 0.793849i 0.917851 + 0.396925i \(0.129923\pi\)
−0.917851 + 0.396925i \(0.870077\pi\)
\(458\) 15.8114i 0.738818i
\(459\) 0 0
\(460\) 8.94427 0.417029
\(461\) −8.94427 −0.416576 −0.208288 0.978068i \(-0.566789\pi\)
−0.208288 + 0.978068i \(0.566789\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 2.82843i 0.131306i
\(465\) 0 0
\(466\) 14.0000 0.648537
\(467\) 12.6491i 0.585331i 0.956215 + 0.292666i \(0.0945422\pi\)
−0.956215 + 0.292666i \(0.905458\pi\)
\(468\) 0 0
\(469\) 10.0000 + 15.8114i 0.461757 + 0.730102i
\(470\) 21.2132i 0.978492i
\(471\) 0 0
\(472\) −4.47214 −0.205847
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) 15.8114i 0.725476i
\(476\) 4.47214 + 7.07107i 0.204980 + 0.324102i
\(477\) 0 0
\(478\) 4.24264i 0.194054i
\(479\) −17.8885 −0.817348 −0.408674 0.912680i \(-0.634009\pi\)
−0.408674 + 0.912680i \(0.634009\pi\)
\(480\) 0 0
\(481\) 44.2719i 2.01862i
\(482\) 25.2982i 1.15230i
\(483\) 0 0
\(484\) −21.0000 −0.954545
\(485\) 0 0
\(486\) 0 0
\(487\) 25.4558i 1.15351i −0.816916 0.576757i \(-0.804318\pi\)
0.816916 0.576757i \(-0.195682\pi\)
\(488\) 9.48683i 0.429449i
\(489\) 0 0
\(490\) −6.70820 + 14.1421i −0.303046 + 0.638877i
\(491\) 19.7990i 0.893516i 0.894655 + 0.446758i \(0.147421\pi\)
−0.894655 + 0.446758i \(0.852579\pi\)
\(492\) 0 0
\(493\) −8.94427 −0.402830
\(494\) 14.1421i 0.636285i
\(495\) 0 0
\(496\) 6.32456i 0.283981i
\(497\) 2.00000 + 3.16228i 0.0897123 + 0.141848i
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 11.1803 0.500000
\(501\) 0 0
\(502\) 0 0
\(503\) 9.48683i 0.422997i −0.977378 0.211498i \(-0.932166\pi\)
0.977378 0.211498i \(-0.0678343\pi\)
\(504\) 0 0
\(505\) −40.0000 −1.77998
\(506\) 22.6274i 1.00591i
\(507\) 0 0
\(508\) 14.1421i 0.627456i
\(509\) 13.4164 0.594672 0.297336 0.954773i \(-0.403902\pi\)
0.297336 + 0.954773i \(0.403902\pi\)
\(510\) 0 0
\(511\) −30.0000 + 18.9737i −1.32712 + 0.839346i
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 15.8114i 0.697410i
\(515\) −20.0000 −0.881305
\(516\) 0 0
\(517\) 53.6656 2.36021
\(518\) −14.0000 22.1359i −0.615125 0.972598i
\(519\) 0 0
\(520\) −10.0000 −0.438529
\(521\) 13.4164 0.587784 0.293892 0.955839i \(-0.405049\pi\)
0.293892 + 0.955839i \(0.405049\pi\)
\(522\) 0 0
\(523\) 17.8885 0.782211 0.391106 0.920346i \(-0.372093\pi\)
0.391106 + 0.920346i \(0.372093\pi\)
\(524\) −17.8885 −0.781465
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) 20.0000 0.871214
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −8.94427 −0.388514
\(531\) 0 0
\(532\) −4.47214 7.07107i −0.193892 0.306570i
\(533\) 20.0000 0.866296
\(534\) 0 0
\(535\) −40.2492 −1.74013
\(536\) 7.07107i 0.305424i
\(537\) 0 0
\(538\) 17.8885 0.771230
\(539\) −35.7771 16.9706i −1.54103 0.730974i
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 6.32456i 0.271663i
\(543\) 0 0
\(544\) 3.16228i 0.135582i
\(545\) −22.3607 −0.957826
\(546\) 0 0
\(547\) 21.2132i 0.907011i 0.891253 + 0.453506i \(0.149827\pi\)
−0.891253 + 0.453506i \(0.850173\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 0 0
\(550\) 28.2843i 1.20605i
\(551\) 8.94427 0.381039
\(552\) 0 0
\(553\) −13.4164 + 8.48528i −0.570524 + 0.360831i
\(554\) 21.2132i 0.901263i
\(555\) 0 0
\(556\) 15.8114i 0.670552i
\(557\) 38.0000 1.61011 0.805056 0.593199i \(-0.202135\pi\)
0.805056 + 0.593199i \(0.202135\pi\)
\(558\) 0 0
\(559\) 6.32456i 0.267500i
\(560\) −5.00000 + 3.16228i −0.211289 + 0.133631i
\(561\) 0 0
\(562\) 12.7279i 0.536895i
\(563\) 18.9737i 0.799645i 0.916593 + 0.399822i \(0.130928\pi\)
−0.916593 + 0.399822i \(0.869072\pi\)
\(564\) 0 0
\(565\) 13.4164 0.564433
\(566\) 0 0
\(567\) 0 0
\(568\) 1.41421i 0.0593391i
\(569\) 24.0416i 1.00788i 0.863739 + 0.503939i \(0.168116\pi\)
−0.863739 + 0.503939i \(0.831884\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 25.2982i 1.05777i
\(573\) 0 0
\(574\) 10.0000 6.32456i 0.417392 0.263982i
\(575\) 20.0000 0.834058
\(576\) 0 0
\(577\) −22.3607 −0.930887 −0.465444 0.885078i \(-0.654105\pi\)
−0.465444 + 0.885078i \(0.654105\pi\)
\(578\) −7.00000 −0.291162
\(579\) 0 0
\(580\) 6.32456i 0.262613i
\(581\) 17.8885 + 28.2843i 0.742142 + 1.17343i
\(582\) 0 0
\(583\) 22.6274i 0.937132i
\(584\) −13.4164 −0.555175
\(585\) 0 0
\(586\) 6.32456i 0.261265i
\(587\) 31.6228i 1.30521i 0.757698 + 0.652606i \(0.226324\pi\)
−0.757698 + 0.652606i \(0.773676\pi\)
\(588\) 0 0
\(589\) −20.0000 −0.824086
\(590\) −10.0000 −0.411693
\(591\) 0 0
\(592\) 9.89949i 0.406867i
\(593\) 9.48683i 0.389578i −0.980845 0.194789i \(-0.937598\pi\)
0.980845 0.194789i \(-0.0624021\pi\)
\(594\) 0 0
\(595\) 10.0000 + 15.8114i 0.409960 + 0.648204i
\(596\) 16.9706i 0.695141i
\(597\) 0 0
\(598\) −17.8885 −0.731517
\(599\) 32.5269i 1.32901i 0.747282 + 0.664507i \(0.231358\pi\)
−0.747282 + 0.664507i \(0.768642\pi\)
\(600\) 0 0
\(601\) 25.2982i 1.03194i −0.856608 0.515968i \(-0.827432\pi\)
0.856608 0.515968i \(-0.172568\pi\)
\(602\) −2.00000 3.16228i −0.0815139 0.128885i
\(603\) 0 0
\(604\) 10.0000 0.406894
\(605\) −46.9574 −1.90909
\(606\) 0 0
\(607\) 8.94427 0.363037 0.181518 0.983388i \(-0.441899\pi\)
0.181518 + 0.983388i \(0.441899\pi\)
\(608\) 3.16228i 0.128247i
\(609\) 0 0
\(610\) 21.2132i 0.858898i
\(611\) 42.4264i 1.71639i
\(612\) 0 0
\(613\) 35.3553i 1.42799i −0.700151 0.713994i \(-0.746884\pi\)
0.700151 0.713994i \(-0.253116\pi\)
\(614\) 26.8328 1.08288
\(615\) 0 0
\(616\) −8.00000 12.6491i −0.322329 0.509647i
\(617\) −38.0000 −1.52982 −0.764911 0.644136i \(-0.777217\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) 0 0
\(619\) 28.4605i 1.14392i −0.820280 0.571962i \(-0.806182\pi\)
0.820280 0.571962i \(-0.193818\pi\)
\(620\) 14.1421i 0.567962i
\(621\) 0 0
\(622\) −8.94427 −0.358633
\(623\) 10.0000 6.32456i 0.400642 0.253388i
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 13.4164 0.536228
\(627\) 0 0
\(628\) 4.47214 0.178458
\(629\) −31.3050 −1.24821
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) −6.00000 −0.238667
\(633\) 0 0
\(634\) −18.0000 −0.714871
\(635\) 31.6228i 1.25491i
\(636\) 0 0
\(637\) 13.4164 28.2843i 0.531577 1.12066i
\(638\) 16.0000 0.633446
\(639\) 0 0
\(640\) −2.23607 −0.0883883
\(641\) 1.41421i 0.0558581i −0.999610 0.0279290i \(-0.991109\pi\)
0.999610 0.0279290i \(-0.00889125\pi\)
\(642\) 0 0
\(643\) 44.7214 1.76364 0.881819 0.471588i \(-0.156319\pi\)
0.881819 + 0.471588i \(0.156319\pi\)
\(644\) −8.94427 + 5.65685i −0.352454 + 0.222911i
\(645\) 0 0
\(646\) −10.0000 −0.393445
\(647\) 34.7851i 1.36754i 0.729697 + 0.683771i \(0.239661\pi\)
−0.729697 + 0.683771i \(0.760339\pi\)
\(648\) 0 0
\(649\) 25.2982i 0.993042i
\(650\) −22.3607 −0.877058
\(651\) 0 0
\(652\) 21.2132i 0.830773i
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) 0 0
\(655\) −40.0000 −1.56293
\(656\) 4.47214 0.174608
\(657\) 0 0
\(658\) 13.4164 + 21.2132i 0.523026 + 0.826977i
\(659\) 16.9706i 0.661079i 0.943792 + 0.330540i \(0.107231\pi\)
−0.943792 + 0.330540i \(0.892769\pi\)
\(660\) 0 0
\(661\) 3.16228i 0.122998i 0.998107 + 0.0614992i \(0.0195882\pi\)
−0.998107 + 0.0614992i \(0.980412\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) 12.6491i 0.490881i
\(665\) −10.0000 15.8114i −0.387783 0.613139i
\(666\) 0 0
\(667\) 11.3137i 0.438069i
\(668\) 9.48683i 0.367057i
\(669\) 0 0
\(670\) 15.8114i 0.610847i
\(671\) 53.6656 2.07174
\(672\) 0 0
\(673\) 19.7990i 0.763195i −0.924329 0.381597i \(-0.875374\pi\)
0.924329 0.381597i \(-0.124626\pi\)
\(674\) 25.4558i 0.980522i
\(675\) 0 0
\(676\) 7.00000 0.269231
\(677\) 37.9473i 1.45843i −0.684282 0.729217i \(-0.739884\pi\)
0.684282 0.729217i \(-0.260116\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 7.07107i 0.271163i
\(681\) 0 0
\(682\) −35.7771 −1.36998
\(683\) 14.0000 0.535695 0.267848 0.963461i \(-0.413688\pi\)
0.267848 + 0.963461i \(0.413688\pi\)
\(684\) 0 0
\(685\) −4.47214 −0.170872
\(686\) −2.23607 18.3848i −0.0853735 0.701934i
\(687\) 0 0
\(688\) 1.41421i 0.0539164i
\(689\) 17.8885 0.681499
\(690\) 0 0
\(691\) 28.4605i 1.08269i −0.840801 0.541344i \(-0.817916\pi\)
0.840801 0.541344i \(-0.182084\pi\)
\(692\) 25.2982i 0.961694i
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 35.3553i 1.34110i
\(696\) 0 0
\(697\) 14.1421i 0.535672i
\(698\) 15.8114i 0.598470i
\(699\) 0 0
\(700\) −11.1803 + 7.07107i −0.422577 + 0.267261i
\(701\) 19.7990i 0.747798i 0.927470 + 0.373899i \(0.121979\pi\)
−0.927470 + 0.373899i \(0.878021\pi\)
\(702\) 0 0
\(703\) 31.3050 1.18069
\(704\) 5.65685i 0.213201i
\(705\) 0 0
\(706\) 3.16228i 0.119014i
\(707\) 40.0000 25.2982i 1.50435 0.951438i
\(708\) 0 0
\(709\) −30.0000 −1.12667 −0.563337 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) 3.16228i 0.118678i
\(711\) 0 0
\(712\) 4.47214 0.167600
\(713\) 25.2982i 0.947426i
\(714\) 0 0
\(715\) 56.5685i 2.11554i
\(716\) 2.82843i 0.105703i
\(717\) 0 0
\(718\) 18.3848i 0.686114i
\(719\) −44.7214 −1.66783 −0.833913 0.551896i \(-0.813904\pi\)
−0.833913 + 0.551896i \(0.813904\pi\)
\(720\) 0 0
\(721\) 20.0000 12.6491i 0.744839 0.471077i
\(722\) −9.00000 −0.334945
\(723\) 0 0
\(724\) 3.16228i 0.117525i
\(725\) 14.1421i 0.525226i
\(726\) 0 0
\(727\) −26.8328 −0.995174 −0.497587 0.867414i \(-0.665780\pi\)
−0.497587 + 0.867414i \(0.665780\pi\)
\(728\) 10.0000 6.32456i 0.370625 0.234404i
\(729\) 0 0
\(730\) −30.0000 −1.11035
\(731\) −4.47214 −0.165408
\(732\) 0 0
\(733\) 40.2492 1.48664 0.743319 0.668937i \(-0.233250\pi\)
0.743319 + 0.668937i \(0.233250\pi\)
\(734\) −13.4164 −0.495209
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) −40.0000 −1.47342
\(738\) 0 0
\(739\) 16.0000 0.588570 0.294285 0.955718i \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) 22.1359i 0.813733i
\(741\) 0 0
\(742\) 8.94427 5.65685i 0.328355 0.207670i
\(743\) 36.0000 1.32071 0.660356 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 0 0
\(745\) 37.9473i 1.39028i
\(746\) 21.2132i 0.776671i
\(747\) 0 0
\(748\) −17.8885 −0.654070
\(749\) 40.2492 25.4558i 1.47067 0.930136i
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 9.48683i 0.345949i
\(753\) 0 0
\(754\) 12.6491i 0.460653i
\(755\) 22.3607 0.813788
\(756\) 0 0
\(757\) 32.5269i 1.18221i 0.806594 + 0.591105i \(0.201308\pi\)
−0.806594 + 0.591105i \(0.798692\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 7.07107i 0.256495i
\(761\) −31.3050 −1.13480 −0.567402 0.823441i \(-0.692051\pi\)
−0.567402 + 0.823441i \(0.692051\pi\)
\(762\) 0 0
\(763\) 22.3607 14.1421i 0.809511 0.511980i
\(764\) 1.41421i 0.0511645i
\(765\) 0 0
\(766\) 9.48683i 0.342773i
\(767\) 20.0000 0.722158
\(768\) 0 0
\(769\) 31.6228i 1.14035i 0.821524 + 0.570173i \(0.193124\pi\)
−0.821524 + 0.570173i \(0.806876\pi\)
\(770\) −17.8885 28.2843i −0.644658 1.01929i
\(771\) 0 0
\(772\) 8.48528i 0.305392i
\(773\) 31.6228i 1.13739i 0.822548 + 0.568696i \(0.192552\pi\)
−0.822548 + 0.568696i \(0.807448\pi\)
\(774\) 0 0
\(775\) 31.6228i 1.13592i
\(776\) 0 0
\(777\) 0 0
\(778\) 16.9706i 0.608424i
\(779\) 14.1421i 0.506695i
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 12.6491i 0.452331i
\(783\) 0 0
\(784\) 3.00000 6.32456i 0.107143 0.225877i
\(785\) 10.0000 0.356915
\(786\) 0 0
\(787\) −26.8328 −0.956487 −0.478243 0.878227i \(-0.658726\pi\)
−0.478243 + 0.878227i \(0.658726\pi\)
\(788\) −12.0000 −0.427482
\(789\) 0 0
\(790\) −13.4164 −0.477334
\(791\) −13.4164 + 8.48528i −0.477033 + 0.301702i
\(792\) 0 0
\(793\) 42.4264i 1.50661i
\(794\) 4.47214 0.158710
\(795\) 0 0
\(796\) 0 0
\(797\) 25.2982i 0.896109i −0.894006 0.448054i \(-0.852117\pi\)
0.894006 0.448054i \(-0.147883\pi\)
\(798\) 0 0
\(799\) 30.0000 1.06132
\(800\) −5.00000 −0.176777
\(801\) 0 0
\(802\) 1.41421i 0.0499376i
\(803\) 75.8947i 2.67826i
\(804\) 0 0
\(805\) −20.0000 + 12.6491i −0.704907 + 0.445823i
\(806\) 28.2843i 0.996271i
\(807\) 0 0
\(808\) 17.8885 0.629317
\(809\) 18.3848i 0.646374i −0.946335 0.323187i \(-0.895246\pi\)
0.946335 0.323187i \(-0.104754\pi\)
\(810\) 0 0
\(811\) 22.1359i 0.777298i −0.921386 0.388649i \(-0.872942\pi\)
0.921386 0.388649i \(-0.127058\pi\)
\(812\) 4.00000 + 6.32456i 0.140372 + 0.221948i
\(813\) 0 0
\(814\) 56.0000 1.96280
\(815\) 47.4342i 1.66155i
\(816\) 0 0
\(817\) 4.47214 0.156460
\(818\) 37.9473i 1.32680i
\(819\) 0 0
\(820\) 10.0000 0.349215
\(821\) 19.7990i 0.690990i −0.938421 0.345495i \(-0.887711\pi\)
0.938421 0.345495i \(-0.112289\pi\)
\(822\) 0 0
\(823\) 28.2843i 0.985928i 0.870050 + 0.492964i \(0.164087\pi\)
−0.870050 + 0.492964i \(0.835913\pi\)
\(824\) 8.94427 0.311588
\(825\) 0 0
\(826\) 10.0000 6.32456i 0.347945 0.220059i
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) 41.1096i 1.42780i 0.700250 + 0.713898i \(0.253072\pi\)
−0.700250 + 0.713898i \(0.746928\pi\)
\(830\) 28.2843i 0.981761i
\(831\) 0 0
\(832\) 4.47214 0.155043
\(833\) −20.0000 9.48683i −0.692959 0.328699i
\(834\) 0 0
\(835\) 21.2132i 0.734113i
\(836\) 17.8885 0.618688
\(837\) 0 0
\(838\) −13.4164 −0.463462
\(839\) 17.8885 0.617581 0.308791 0.951130i \(-0.400076\pi\)
0.308791 + 0.951130i \(0.400076\pi\)
\(840\) 0 0
\(841\) 21.0000 0.724138
\(842\) −38.0000 −1.30957
\(843\) 0 0
\(844\) −20.0000 −0.688428
\(845\) 15.6525 0.538462
\(846\) 0 0
\(847\) 46.9574 29.6985i 1.61348 1.02045i
\(848\) 4.00000 0.137361
\(849\) 0 0
\(850\) 15.8114i 0.542326i
\(851\) 39.5980i 1.35740i
\(852\) 0 0
\(853\) −49.1935 −1.68435 −0.842177 0.539202i \(-0.818726\pi\)
−0.842177 + 0.539202i \(0.818726\pi\)
\(854\) 13.4164 + 21.2132i 0.459100 + 0.725901i
\(855\) 0 0
\(856\) 18.0000 0.615227
\(857\) 41.1096i 1.40428i 0.712040 + 0.702139i \(0.247771\pi\)
−0.712040 + 0.702139i \(0.752229\pi\)
\(858\) 0 0
\(859\) 34.7851i 1.18685i 0.804889 + 0.593425i \(0.202225\pi\)
−0.804889 + 0.593425i \(0.797775\pi\)
\(860\) 3.16228i 0.107833i
\(861\) 0 0
\(862\) 15.5563i 0.529851i
\(863\) 36.0000 1.22545 0.612727 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 0 0
\(865\) 56.5685i 1.92339i
\(866\) 26.8328 0.911816
\(867\) 0 0
\(868\) −8.94427 14.1421i −0.303588 0.480015i
\(869\) 33.9411i 1.15137i
\(870\) 0 0
\(871\) 31.6228i 1.07150i
\(872\) 10.0000 0.338643
\(873\) 0 0
\(874\) 12.6491i 0.427863i
\(875\) −25.0000 + 15.8114i −0.845154 + 0.534522i
\(876\) 0 0
\(877\) 21.2132i 0.716319i 0.933660 + 0.358159i \(0.116596\pi\)
−0.933660 + 0.358159i \(0.883404\pi\)
\(878\) 6.32456i 0.213443i
\(879\) 0 0
\(880\) 12.6491i 0.426401i
\(881\) 4.47214 0.150670 0.0753350 0.997158i \(-0.475997\pi\)
0.0753350 + 0.997158i \(0.475997\pi\)
\(882\) 0 0
\(883\) 12.7279i 0.428329i −0.976798 0.214164i \(-0.931297\pi\)
0.976798 0.214164i \(-0.0687028\pi\)
\(884\) 14.1421i 0.475651i
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) 9.48683i 0.318537i −0.987235 0.159268i \(-0.949086\pi\)
0.987235 0.159268i \(-0.0509135\pi\)
\(888\) 0 0
\(889\) −20.0000 31.6228i −0.670778 1.06059i
\(890\) 10.0000 0.335201
\(891\) 0 0
\(892\) 8.94427 0.299476
\(893\) −30.0000 −1.00391
\(894\) 0 0
\(895\) 6.32456i 0.211407i
\(896\) 2.23607 1.41421i 0.0747018 0.0472456i
\(897\) 0 0
\(898\) 9.89949i 0.330350i
\(899\) 17.8885 0.596616
\(900\) 0 0
\(901\) 12.6491i 0.421403i
\(902\) 25.2982i 0.842339i
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 7.07107i 0.235050i
\(906\) 0 0
\(907\) 46.6690i 1.54962i −0.632194 0.774810i \(-0.717845\pi\)
0.632194 0.774810i \(-0.282155\pi\)
\(908\) 18.9737i 0.629663i
\(909\) 0 0
\(910\) 22.3607 14.1421i 0.741249 0.468807i
\(911\) 43.8406i 1.45250i −0.687428 0.726252i \(-0.741260\pi\)
0.687428 0.726252i \(-0.258740\pi\)
\(912\) 0 0
\(913\) −71.5542 −2.36810
\(914\) 16.9706i 0.561336i
\(915\) 0 0
\(916\) 15.8114i 0.522423i
\(917\) 40.0000 25.2982i 1.32092 0.835421i
\(918\) 0 0
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) −8.94427 −0.294884
\(921\) 0 0
\(922\) 8.94427 0.294564
\(923\) 6.32456i 0.208175i
\(924\) 0 0
\(925\) 49.4975i 1.62747i
\(926\) 0 0
\(927\) 0 0
\(928\) 2.82843i 0.0928477i
\(929\) 22.3607 0.733630 0.366815 0.930294i \(-0.380448\pi\)
0.366815 + 0.930294i \(0.380448\pi\)
\(930\) 0 0
\(931\) 20.0000 + 9.48683i 0.655474 + 0.310918i
\(932\) −14.0000 −0.458585
\(933\) 0 0
\(934\) 12.6491i 0.413892i
\(935\) −40.0000 −1.30814
\(936\) 0 0
\(937\) −53.6656 −1.75318 −0.876590 0.481238i \(-0.840187\pi\)
−0.876590 + 0.481238i \(0.840187\pi\)
\(938\) −10.0000 15.8114i −0.326512 0.516260i
\(939\) 0 0
\(940\) 21.2132i 0.691898i
\(941\) −26.8328 −0.874725 −0.437362 0.899285i \(-0.644087\pi\)
−0.437362 + 0.899285i \(0.644087\pi\)
\(942\) 0 0
\(943\) 17.8885 0.582531
\(944\) 4.47214 0.145556
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) −42.0000 −1.36482 −0.682408 0.730971i \(-0.739067\pi\)
−0.682408 + 0.730971i \(0.739067\pi\)
\(948\) 0 0
\(949\) 60.0000 1.94768
\(950\) 15.8114i 0.512989i
\(951\) 0 0
\(952\) −4.47214 7.07107i −0.144943 0.229175i
\(953\) −54.0000 −1.74923 −0.874616 0.484817i \(-0.838886\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(954\) 0 0
\(955\) 3.16228i 0.102329i
\(956\) 4.24264i 0.137217i
\(957\) 0 0
\(958\) 17.8885 0.577953
\(959\) 4.47214 2.82843i 0.144413 0.0913347i
\(960\) 0 0
\(961\) −9.00000 −0.290323
\(962\) 44.2719i 1.42738i
\(963\) 0 0
\(964\) 25.2982i 0.814801i
\(965\) 18.9737i 0.610784i
\(966\) 0 0
\(967\) 42.4264i 1.36434i 0.731193 + 0.682171i \(0.238964\pi\)
−0.731193 + 0.682171i \(0.761036\pi\)
\(968\) 21.0000 0.674966
\(969\) 0 0
\(970\) 0 0
\(971\) 35.7771 1.14814 0.574071 0.818806i \(-0.305363\pi\)
0.574071 + 0.818806i \(0.305363\pi\)
\(972\) 0 0
\(973\) 22.3607 + 35.3553i 0.716850 + 1.13344i
\(974\) 25.4558i 0.815658i
\(975\) 0 0
\(976\) 9.48683i 0.303666i
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) 25.2982i 0.808535i
\(980\) 6.70820 14.1421i 0.214286 0.451754i
\(981\) 0 0
\(982\) 19.7990i 0.631811i
\(983\) 9.48683i 0.302583i −0.988489 0.151291i \(-0.951657\pi\)
0.988489 0.151291i \(-0.0483432\pi\)
\(984\) 0 0
\(985\) −26.8328 −0.854965
\(986\) 8.94427 0.284844
\(987\) 0 0
\(988\) 14.1421i 0.449921i
\(989\) 5.65685i 0.179878i
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 6.32456i 0.200805i
\(993\) 0 0
\(994\) −2.00000 3.16228i −0.0634361 0.100301i
\(995\) 0 0
\(996\) 0 0
\(997\) 13.4164 0.424902 0.212451 0.977172i \(-0.431855\pi\)
0.212451 + 0.977172i \(0.431855\pi\)
\(998\) 20.0000 0.633089
\(999\) 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 630.2.d.b.629.4 yes 4
3.2 odd 2 630.2.d.c.629.2 yes 4
4.3 odd 2 5040.2.k.b.1889.3 4
5.2 odd 4 3150.2.b.b.251.1 8
5.3 odd 4 3150.2.b.b.251.8 8
5.4 even 2 630.2.d.c.629.3 yes 4
7.6 odd 2 inner 630.2.d.b.629.2 yes 4
12.11 even 2 5040.2.k.c.1889.1 4
15.2 even 4 3150.2.b.b.251.5 8
15.8 even 4 3150.2.b.b.251.4 8
15.14 odd 2 inner 630.2.d.b.629.1 4
20.19 odd 2 5040.2.k.c.1889.4 4
21.20 even 2 630.2.d.c.629.4 yes 4
28.27 even 2 5040.2.k.b.1889.1 4
35.13 even 4 3150.2.b.b.251.7 8
35.27 even 4 3150.2.b.b.251.2 8
35.34 odd 2 630.2.d.c.629.1 yes 4
60.59 even 2 5040.2.k.b.1889.2 4
84.83 odd 2 5040.2.k.c.1889.3 4
105.62 odd 4 3150.2.b.b.251.6 8
105.83 odd 4 3150.2.b.b.251.3 8
105.104 even 2 inner 630.2.d.b.629.3 yes 4
140.139 even 2 5040.2.k.c.1889.2 4
420.419 odd 2 5040.2.k.b.1889.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
630.2.d.b.629.1 4 15.14 odd 2 inner
630.2.d.b.629.2 yes 4 7.6 odd 2 inner
630.2.d.b.629.3 yes 4 105.104 even 2 inner
630.2.d.b.629.4 yes 4 1.1 even 1 trivial
630.2.d.c.629.1 yes 4 35.34 odd 2
630.2.d.c.629.2 yes 4 3.2 odd 2
630.2.d.c.629.3 yes 4 5.4 even 2
630.2.d.c.629.4 yes 4 21.20 even 2
3150.2.b.b.251.1 8 5.2 odd 4
3150.2.b.b.251.2 8 35.27 even 4
3150.2.b.b.251.3 8 105.83 odd 4
3150.2.b.b.251.4 8 15.8 even 4
3150.2.b.b.251.5 8 15.2 even 4
3150.2.b.b.251.6 8 105.62 odd 4
3150.2.b.b.251.7 8 35.13 even 4
3150.2.b.b.251.8 8 5.3 odd 4
5040.2.k.b.1889.1 4 28.27 even 2
5040.2.k.b.1889.2 4 60.59 even 2
5040.2.k.b.1889.3 4 4.3 odd 2
5040.2.k.b.1889.4 4 420.419 odd 2
5040.2.k.c.1889.1 4 12.11 even 2
5040.2.k.c.1889.2 4 140.139 even 2
5040.2.k.c.1889.3 4 84.83 odd 2
5040.2.k.c.1889.4 4 20.19 odd 2