Properties

Label 2025.2.b.o.649.5
Level $2025$
Weight $2$
Character 2025.649
Analytic conductor $16.170$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2025,2,Mod(649,2025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2025.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2025.b (of order \(2\), degree \(1\), not 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: \(16.1697064093\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.34810603776.6
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 12x^{6} + 42x^{4} + 49x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 225)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.5
Root \(0.473255i\) of defining polynomial
Character \(\chi\) \(=\) 2025.649
Dual form 2025.2.b.o.649.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.473255i q^{2} +1.77603 q^{4} -2.56305i q^{7} +1.78702i q^{8} +O(q^{10})\) \(q+0.473255i q^{2} +1.77603 q^{4} -2.56305i q^{7} +1.78702i q^{8} +6.16860 q^{11} +2.13230i q^{13} +1.21298 q^{14} +2.70634 q^{16} +3.16860i q^{17} -0.356267 q^{19} +2.91932i q^{22} +4.21298i q^{23} -1.00912 q^{26} -4.55206i q^{28} +1.68623 q^{29} -8.25840 q^{31} +4.85484i q^{32} -1.49956 q^{34} -3.63274i q^{37} -0.168605i q^{38} +2.73353 q^{41} -7.67817i q^{43} +10.9556 q^{44} -1.99381 q^{46} -11.4289i q^{47} +0.430757 q^{49} +3.78702i q^{52} +9.43507i q^{53} +4.58024 q^{56} +0.798017i q^{58} +10.2159 q^{59} -0.0109932 q^{61} -3.90833i q^{62} +3.11511 q^{64} -0.982817i q^{67} +5.62754i q^{68} +6.43507 q^{71} +6.61467i q^{73} +1.71921 q^{74} -0.632740 q^{76} -15.8105i q^{77} +9.47138 q^{79} +1.29366i q^{82} +10.4198i q^{83} +3.63373 q^{86} +11.0234i q^{88} -6.26940 q^{89} +5.46519 q^{91} +7.48237i q^{92} +5.40877 q^{94} -7.20679i q^{97} +0.203858i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 2 q^{11} + 6 q^{14} + 8 q^{16} - 4 q^{19} + 20 q^{26} + 2 q^{29} - 8 q^{31} - 18 q^{34} + 10 q^{41} + 44 q^{44} + 6 q^{49} + 60 q^{56} + 34 q^{59} - 26 q^{61} - 38 q^{64} + 16 q^{71} + 80 q^{74} + 22 q^{76} + 14 q^{79} + 68 q^{86} - 18 q^{89} - 34 q^{91} - 6 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(1702\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.473255i 0.334641i 0.985903 + 0.167321i \(0.0535115\pi\)
−0.985903 + 0.167321i \(0.946488\pi\)
\(3\) 0 0
\(4\) 1.77603 0.888015
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.56305i − 0.968743i −0.874862 0.484372i \(-0.839048\pi\)
0.874862 0.484372i \(-0.160952\pi\)
\(8\) 1.78702i 0.631808i
\(9\) 0 0
\(10\) 0 0
\(11\) 6.16860 1.85990 0.929952 0.367681i \(-0.119848\pi\)
0.929952 + 0.367681i \(0.119848\pi\)
\(12\) 0 0
\(13\) 2.13230i 0.591393i 0.955282 + 0.295696i \(0.0955517\pi\)
−0.955282 + 0.295696i \(0.904448\pi\)
\(14\) 1.21298 0.324182
\(15\) 0 0
\(16\) 2.70634 0.676586
\(17\) 3.16860i 0.768500i 0.923229 + 0.384250i \(0.125540\pi\)
−0.923229 + 0.384250i \(0.874460\pi\)
\(18\) 0 0
\(19\) −0.356267 −0.0817332 −0.0408666 0.999165i \(-0.513012\pi\)
−0.0408666 + 0.999165i \(0.513012\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.91932i 0.622401i
\(23\) 4.21298i 0.878466i 0.898373 + 0.439233i \(0.144750\pi\)
−0.898373 + 0.439233i \(0.855250\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.00912 −0.197905
\(27\) 0 0
\(28\) − 4.55206i − 0.860259i
\(29\) 1.68623 0.313125 0.156563 0.987668i \(-0.449959\pi\)
0.156563 + 0.987668i \(0.449959\pi\)
\(30\) 0 0
\(31\) −8.25840 −1.48325 −0.741627 0.670813i \(-0.765945\pi\)
−0.741627 + 0.670813i \(0.765945\pi\)
\(32\) 4.85484i 0.858222i
\(33\) 0 0
\(34\) −1.49956 −0.257172
\(35\) 0 0
\(36\) 0 0
\(37\) − 3.63274i − 0.597219i −0.954375 0.298609i \(-0.903477\pi\)
0.954375 0.298609i \(-0.0965228\pi\)
\(38\) − 0.168605i − 0.0273513i
\(39\) 0 0
\(40\) 0 0
\(41\) 2.73353 0.426906 0.213453 0.976953i \(-0.431529\pi\)
0.213453 + 0.976953i \(0.431529\pi\)
\(42\) 0 0
\(43\) − 7.67817i − 1.17091i −0.810705 0.585455i \(-0.800916\pi\)
0.810705 0.585455i \(-0.199084\pi\)
\(44\) 10.9556 1.65162
\(45\) 0 0
\(46\) −1.99381 −0.293971
\(47\) − 11.4289i − 1.66707i −0.552464 0.833537i \(-0.686312\pi\)
0.552464 0.833537i \(-0.313688\pi\)
\(48\) 0 0
\(49\) 0.430757 0.0615367
\(50\) 0 0
\(51\) 0 0
\(52\) 3.78702i 0.525166i
\(53\) 9.43507i 1.29601i 0.761637 + 0.648003i \(0.224396\pi\)
−0.761637 + 0.648003i \(0.775604\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.58024 0.612060
\(57\) 0 0
\(58\) 0.798017i 0.104785i
\(59\) 10.2159 1.33000 0.664999 0.746844i \(-0.268432\pi\)
0.664999 + 0.746844i \(0.268432\pi\)
\(60\) 0 0
\(61\) −0.0109932 −0.00140753 −0.000703767 1.00000i \(-0.500224\pi\)
−0.000703767 1.00000i \(0.500224\pi\)
\(62\) − 3.90833i − 0.496358i
\(63\) 0 0
\(64\) 3.11511 0.389389
\(65\) 0 0
\(66\) 0 0
\(67\) − 0.982817i − 0.120070i −0.998196 0.0600351i \(-0.980879\pi\)
0.998196 0.0600351i \(-0.0191213\pi\)
\(68\) 5.62754i 0.682439i
\(69\) 0 0
\(70\) 0 0
\(71\) 6.43507 0.763703 0.381851 0.924224i \(-0.375287\pi\)
0.381851 + 0.924224i \(0.375287\pi\)
\(72\) 0 0
\(73\) 6.61467i 0.774189i 0.922040 + 0.387094i \(0.126521\pi\)
−0.922040 + 0.387094i \(0.873479\pi\)
\(74\) 1.71921 0.199854
\(75\) 0 0
\(76\) −0.632740 −0.0725803
\(77\) − 15.8105i − 1.80177i
\(78\) 0 0
\(79\) 9.47138 1.06561 0.532807 0.846237i \(-0.321137\pi\)
0.532807 + 0.846237i \(0.321137\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1.29366i 0.142860i
\(83\) 10.4198i 1.14372i 0.820352 + 0.571859i \(0.193777\pi\)
−0.820352 + 0.571859i \(0.806223\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 3.63373 0.391835
\(87\) 0 0
\(88\) 11.0234i 1.17510i
\(89\) −6.26940 −0.664555 −0.332277 0.943182i \(-0.607817\pi\)
−0.332277 + 0.943182i \(0.607817\pi\)
\(90\) 0 0
\(91\) 5.46519 0.572908
\(92\) 7.48237i 0.780091i
\(93\) 0 0
\(94\) 5.40877 0.557872
\(95\) 0 0
\(96\) 0 0
\(97\) − 7.20679i − 0.731738i −0.930666 0.365869i \(-0.880772\pi\)
0.930666 0.365869i \(-0.119228\pi\)
\(98\) 0.203858i 0.0205927i
\(99\) 0 0
\(100\) 0 0
\(101\) −6.97094 −0.693634 −0.346817 0.937933i \(-0.612738\pi\)
−0.346817 + 0.937933i \(0.612738\pi\)
\(102\) 0 0
\(103\) − 6.11511i − 0.602540i −0.953539 0.301270i \(-0.902589\pi\)
0.953539 0.301270i \(-0.0974106\pi\)
\(104\) −3.81046 −0.373647
\(105\) 0 0
\(106\) −4.46519 −0.433698
\(107\) − 14.5349i − 1.40514i −0.711615 0.702570i \(-0.752036\pi\)
0.711615 0.702570i \(-0.247964\pi\)
\(108\) 0 0
\(109\) −1.90214 −0.182192 −0.0910958 0.995842i \(-0.529037\pi\)
−0.0910958 + 0.995842i \(0.529037\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 6.93650i − 0.655438i
\(113\) 6.57925i 0.618924i 0.950912 + 0.309462i \(0.100149\pi\)
−0.950912 + 0.309462i \(0.899851\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2.99480 0.278060
\(117\) 0 0
\(118\) 4.83472i 0.445072i
\(119\) 8.12130 0.744479
\(120\) 0 0
\(121\) 27.0517 2.45924
\(122\) − 0.00520257i 0 0.000471019i
\(123\) 0 0
\(124\) −14.6672 −1.31715
\(125\) 0 0
\(126\) 0 0
\(127\) 9.25840i 0.821550i 0.911737 + 0.410775i \(0.134742\pi\)
−0.911737 + 0.410775i \(0.865258\pi\)
\(128\) 11.1839i 0.988528i
\(129\) 0 0
\(130\) 0 0
\(131\) −0.269397 −0.0235373 −0.0117687 0.999931i \(-0.503746\pi\)
−0.0117687 + 0.999931i \(0.503746\pi\)
\(132\) 0 0
\(133\) 0.913130i 0.0791784i
\(134\) 0.465123 0.0401805
\(135\) 0 0
\(136\) −5.66237 −0.485544
\(137\) 3.47618i 0.296990i 0.988913 + 0.148495i \(0.0474430\pi\)
−0.988913 + 0.148495i \(0.952557\pi\)
\(138\) 0 0
\(139\) −14.7479 −1.25090 −0.625448 0.780266i \(-0.715084\pi\)
−0.625448 + 0.780266i \(0.715084\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 3.04543i 0.255567i
\(143\) 13.1533i 1.09993i
\(144\) 0 0
\(145\) 0 0
\(146\) −3.13042 −0.259076
\(147\) 0 0
\(148\) − 6.45186i − 0.530339i
\(149\) −10.1533 −0.831790 −0.415895 0.909413i \(-0.636532\pi\)
−0.415895 + 0.909413i \(0.636532\pi\)
\(150\) 0 0
\(151\) −10.3162 −0.839521 −0.419761 0.907635i \(-0.637886\pi\)
−0.419761 + 0.907635i \(0.637886\pi\)
\(152\) − 0.636657i − 0.0516397i
\(153\) 0 0
\(154\) 7.48237 0.602947
\(155\) 0 0
\(156\) 0 0
\(157\) − 1.06261i − 0.0848055i −0.999101 0.0424028i \(-0.986499\pi\)
0.999101 0.0424028i \(-0.0135013\pi\)
\(158\) 4.48237i 0.356598i
\(159\) 0 0
\(160\) 0 0
\(161\) 10.7981 0.851008
\(162\) 0 0
\(163\) − 17.1386i − 1.34240i −0.741278 0.671198i \(-0.765780\pi\)
0.741278 0.671198i \(-0.234220\pi\)
\(164\) 4.85484 0.379099
\(165\) 0 0
\(166\) −4.93120 −0.382735
\(167\) − 4.37345i − 0.338428i −0.985579 0.169214i \(-0.945877\pi\)
0.985579 0.169214i \(-0.0541229\pi\)
\(168\) 0 0
\(169\) 8.45331 0.650255
\(170\) 0 0
\(171\) 0 0
\(172\) − 13.6367i − 1.03979i
\(173\) 14.6601i 1.11459i 0.830316 + 0.557293i \(0.188160\pi\)
−0.830316 + 0.557293i \(0.811840\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 16.6944 1.25838
\(177\) 0 0
\(178\) − 2.96702i − 0.222388i
\(179\) −6.87014 −0.513499 −0.256749 0.966478i \(-0.582651\pi\)
−0.256749 + 0.966478i \(0.582651\pi\)
\(180\) 0 0
\(181\) −10.9709 −0.815463 −0.407732 0.913102i \(-0.633680\pi\)
−0.407732 + 0.913102i \(0.633680\pi\)
\(182\) 2.58643i 0.191719i
\(183\) 0 0
\(184\) −7.52869 −0.555022
\(185\) 0 0
\(186\) 0 0
\(187\) 19.5459i 1.42934i
\(188\) − 20.2980i − 1.48039i
\(189\) 0 0
\(190\) 0 0
\(191\) −13.7325 −0.993652 −0.496826 0.867850i \(-0.665501\pi\)
−0.496826 + 0.867850i \(0.665501\pi\)
\(192\) 0 0
\(193\) − 0.482374i − 0.0347220i −0.999849 0.0173610i \(-0.994474\pi\)
0.999849 0.0173610i \(-0.00552646\pi\)
\(194\) 3.41064 0.244870
\(195\) 0 0
\(196\) 0.765037 0.0546455
\(197\) 5.53488i 0.394344i 0.980369 + 0.197172i \(0.0631757\pi\)
−0.980369 + 0.197172i \(0.936824\pi\)
\(198\) 0 0
\(199\) −17.4590 −1.23764 −0.618818 0.785534i \(-0.712388\pi\)
−0.618818 + 0.785534i \(0.712388\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 3.29903i − 0.232119i
\(203\) − 4.32190i − 0.303338i
\(204\) 0 0
\(205\) 0 0
\(206\) 2.89401 0.201635
\(207\) 0 0
\(208\) 5.77073i 0.400128i
\(209\) −2.19767 −0.152016
\(210\) 0 0
\(211\) −1.63666 −0.112672 −0.0563360 0.998412i \(-0.517942\pi\)
−0.0563360 + 0.998412i \(0.517942\pi\)
\(212\) 16.7570i 1.15087i
\(213\) 0 0
\(214\) 6.87870 0.470218
\(215\) 0 0
\(216\) 0 0
\(217\) 21.1667i 1.43689i
\(218\) − 0.900195i − 0.0609689i
\(219\) 0 0
\(220\) 0 0
\(221\) −6.75641 −0.454485
\(222\) 0 0
\(223\) 7.74785i 0.518835i 0.965765 + 0.259417i \(0.0835305\pi\)
−0.965765 + 0.259417i \(0.916469\pi\)
\(224\) 12.4432 0.831397
\(225\) 0 0
\(226\) −3.11366 −0.207118
\(227\) − 11.2603i − 0.747371i −0.927555 0.373685i \(-0.878094\pi\)
0.927555 0.373685i \(-0.121906\pi\)
\(228\) 0 0
\(229\) 10.4776 0.692377 0.346189 0.938165i \(-0.387476\pi\)
0.346189 + 0.938165i \(0.387476\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 3.01333i 0.197835i
\(233\) 2.90214i 0.190125i 0.995471 + 0.0950627i \(0.0303051\pi\)
−0.995471 + 0.0950627i \(0.969695\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 18.1438 1.18106
\(237\) 0 0
\(238\) 3.84344i 0.249133i
\(239\) 16.3545 1.05788 0.528941 0.848659i \(-0.322589\pi\)
0.528941 + 0.848659i \(0.322589\pi\)
\(240\) 0 0
\(241\) 17.5239 1.12881 0.564406 0.825497i \(-0.309105\pi\)
0.564406 + 0.825497i \(0.309105\pi\)
\(242\) 12.8023i 0.822965i
\(243\) 0 0
\(244\) −0.0195242 −0.00124991
\(245\) 0 0
\(246\) 0 0
\(247\) − 0.759666i − 0.0483364i
\(248\) − 14.7580i − 0.937131i
\(249\) 0 0
\(250\) 0 0
\(251\) −8.46999 −0.534621 −0.267311 0.963610i \(-0.586135\pi\)
−0.267311 + 0.963610i \(0.586135\pi\)
\(252\) 0 0
\(253\) 25.9882i 1.63386i
\(254\) −4.38158 −0.274925
\(255\) 0 0
\(256\) 0.937390 0.0585869
\(257\) − 2.87251i − 0.179182i −0.995979 0.0895910i \(-0.971444\pi\)
0.995979 0.0895910i \(-0.0285560\pi\)
\(258\) 0 0
\(259\) −9.31091 −0.578552
\(260\) 0 0
\(261\) 0 0
\(262\) − 0.127493i − 0.00787656i
\(263\) − 25.6239i − 1.58003i −0.613084 0.790017i \(-0.710071\pi\)
0.613084 0.790017i \(-0.289929\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −0.432143 −0.0264964
\(267\) 0 0
\(268\) − 1.74551i − 0.106624i
\(269\) 0.337210 0.0205600 0.0102800 0.999947i \(-0.496728\pi\)
0.0102800 + 0.999947i \(0.496728\pi\)
\(270\) 0 0
\(271\) 21.5927 1.31166 0.655831 0.754908i \(-0.272318\pi\)
0.655831 + 0.754908i \(0.272318\pi\)
\(272\) 8.57533i 0.519956i
\(273\) 0 0
\(274\) −1.64512 −0.0993853
\(275\) 0 0
\(276\) 0 0
\(277\) − 24.1338i − 1.45006i −0.688719 0.725028i \(-0.741827\pi\)
0.688719 0.725028i \(-0.258173\pi\)
\(278\) − 6.97949i − 0.418602i
\(279\) 0 0
\(280\) 0 0
\(281\) −3.36726 −0.200874 −0.100437 0.994943i \(-0.532024\pi\)
−0.100437 + 0.994943i \(0.532024\pi\)
\(282\) 0 0
\(283\) 21.8497i 1.29883i 0.760434 + 0.649415i \(0.224986\pi\)
−0.760434 + 0.649415i \(0.775014\pi\)
\(284\) 11.4289 0.678179
\(285\) 0 0
\(286\) −6.22486 −0.368083
\(287\) − 7.00619i − 0.413562i
\(288\) 0 0
\(289\) 6.95994 0.409408
\(290\) 0 0
\(291\) 0 0
\(292\) 11.7479i 0.687491i
\(293\) − 13.7540i − 0.803520i −0.915745 0.401760i \(-0.868399\pi\)
0.915745 0.401760i \(-0.131601\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 6.49179 0.377328
\(297\) 0 0
\(298\) − 4.80509i − 0.278352i
\(299\) −8.98332 −0.519519
\(300\) 0 0
\(301\) −19.6796 −1.13431
\(302\) − 4.88219i − 0.280939i
\(303\) 0 0
\(304\) −0.964180 −0.0552995
\(305\) 0 0
\(306\) 0 0
\(307\) − 34.2183i − 1.95294i −0.215644 0.976472i \(-0.569185\pi\)
0.215644 0.976472i \(-0.430815\pi\)
\(308\) − 28.0799i − 1.60000i
\(309\) 0 0
\(310\) 0 0
\(311\) −23.0397 −1.30646 −0.653232 0.757158i \(-0.726587\pi\)
−0.653232 + 0.757158i \(0.726587\pi\)
\(312\) 0 0
\(313\) − 3.59130i − 0.202992i −0.994836 0.101496i \(-0.967637\pi\)
0.994836 0.101496i \(-0.0323629\pi\)
\(314\) 0.502885 0.0283794
\(315\) 0 0
\(316\) 16.8215 0.946281
\(317\) 13.1807i 0.740299i 0.928972 + 0.370150i \(0.120694\pi\)
−0.928972 + 0.370150i \(0.879306\pi\)
\(318\) 0 0
\(319\) 10.4017 0.582383
\(320\) 0 0
\(321\) 0 0
\(322\) 5.11024i 0.284783i
\(323\) − 1.12887i − 0.0628119i
\(324\) 0 0
\(325\) 0 0
\(326\) 8.11090 0.449221
\(327\) 0 0
\(328\) 4.88489i 0.269723i
\(329\) −29.2928 −1.61497
\(330\) 0 0
\(331\) 1.18253 0.0649976 0.0324988 0.999472i \(-0.489653\pi\)
0.0324988 + 0.999472i \(0.489653\pi\)
\(332\) 18.5058i 1.01564i
\(333\) 0 0
\(334\) 2.06975 0.113252
\(335\) 0 0
\(336\) 0 0
\(337\) 24.7995i 1.35091i 0.737400 + 0.675457i \(0.236053\pi\)
−0.737400 + 0.675457i \(0.763947\pi\)
\(338\) 4.00057i 0.217602i
\(339\) 0 0
\(340\) 0 0
\(341\) −50.9428 −2.75871
\(342\) 0 0
\(343\) − 19.0454i − 1.02836i
\(344\) 13.7211 0.739790
\(345\) 0 0
\(346\) −6.93796 −0.372987
\(347\) 22.1692i 1.19010i 0.803687 + 0.595052i \(0.202868\pi\)
−0.803687 + 0.595052i \(0.797132\pi\)
\(348\) 0 0
\(349\) −14.9185 −0.798569 −0.399285 0.916827i \(-0.630741\pi\)
−0.399285 + 0.916827i \(0.630741\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 29.9476i 1.59621i
\(353\) − 16.9145i − 0.900269i −0.892961 0.450134i \(-0.851376\pi\)
0.892961 0.450134i \(-0.148624\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −11.1346 −0.590135
\(357\) 0 0
\(358\) − 3.25133i − 0.171838i
\(359\) 0.636657 0.0336015 0.0168007 0.999859i \(-0.494652\pi\)
0.0168007 + 0.999859i \(0.494652\pi\)
\(360\) 0 0
\(361\) −18.8731 −0.993320
\(362\) − 5.19205i − 0.272888i
\(363\) 0 0
\(364\) 9.70634 0.508751
\(365\) 0 0
\(366\) 0 0
\(367\) − 20.0979i − 1.04910i −0.851379 0.524552i \(-0.824233\pi\)
0.851379 0.524552i \(-0.175767\pi\)
\(368\) 11.4018i 0.594358i
\(369\) 0 0
\(370\) 0 0
\(371\) 24.1826 1.25550
\(372\) 0 0
\(373\) − 19.6429i − 1.01707i −0.861041 0.508536i \(-0.830187\pi\)
0.861041 0.508536i \(-0.169813\pi\)
\(374\) −9.25017 −0.478315
\(375\) 0 0
\(376\) 20.4237 1.05327
\(377\) 3.59555i 0.185180i
\(378\) 0 0
\(379\) 7.94219 0.407963 0.203982 0.978975i \(-0.434612\pi\)
0.203982 + 0.978975i \(0.434612\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 6.49899i − 0.332517i
\(383\) 31.1888i 1.59367i 0.604195 + 0.796836i \(0.293495\pi\)
−0.604195 + 0.796836i \(0.706505\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0.228285 0.0116194
\(387\) 0 0
\(388\) − 12.7995i − 0.649795i
\(389\) −31.4494 −1.59455 −0.797274 0.603618i \(-0.793725\pi\)
−0.797274 + 0.603618i \(0.793725\pi\)
\(390\) 0 0
\(391\) −13.3493 −0.675101
\(392\) 0.769772i 0.0388794i
\(393\) 0 0
\(394\) −2.61941 −0.131964
\(395\) 0 0
\(396\) 0 0
\(397\) − 17.7174i − 0.889211i −0.895726 0.444606i \(-0.853344\pi\)
0.895726 0.444606i \(-0.146656\pi\)
\(398\) − 8.26255i − 0.414164i
\(399\) 0 0
\(400\) 0 0
\(401\) 7.14247 0.356678 0.178339 0.983969i \(-0.442928\pi\)
0.178339 + 0.983969i \(0.442928\pi\)
\(402\) 0 0
\(403\) − 17.6094i − 0.877185i
\(404\) −12.3806 −0.615958
\(405\) 0 0
\(406\) 2.04536 0.101509
\(407\) − 22.4089i − 1.11077i
\(408\) 0 0
\(409\) −24.7518 −1.22390 −0.611948 0.790898i \(-0.709614\pi\)
−0.611948 + 0.790898i \(0.709614\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 10.8606i − 0.535065i
\(413\) − 26.1839i − 1.28843i
\(414\) 0 0
\(415\) 0 0
\(416\) −10.3520 −0.507546
\(417\) 0 0
\(418\) − 1.04006i − 0.0508708i
\(419\) −10.6591 −0.520732 −0.260366 0.965510i \(-0.583843\pi\)
−0.260366 + 0.965510i \(0.583843\pi\)
\(420\) 0 0
\(421\) −8.17861 −0.398601 −0.199301 0.979938i \(-0.563867\pi\)
−0.199301 + 0.979938i \(0.563867\pi\)
\(422\) − 0.774555i − 0.0377048i
\(423\) 0 0
\(424\) −16.8607 −0.818828
\(425\) 0 0
\(426\) 0 0
\(427\) 0.0281761i 0.00136354i
\(428\) − 25.8144i − 1.24779i
\(429\) 0 0
\(430\) 0 0
\(431\) −1.67248 −0.0805604 −0.0402802 0.999188i \(-0.512825\pi\)
−0.0402802 + 0.999188i \(0.512825\pi\)
\(432\) 0 0
\(433\) − 9.95994i − 0.478644i −0.970940 0.239322i \(-0.923075\pi\)
0.970940 0.239322i \(-0.0769252\pi\)
\(434\) −10.0173 −0.480843
\(435\) 0 0
\(436\) −3.37825 −0.161789
\(437\) − 1.50094i − 0.0717998i
\(438\) 0 0
\(439\) −12.8158 −0.611663 −0.305832 0.952086i \(-0.598934\pi\)
−0.305832 + 0.952086i \(0.598934\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 3.19750i − 0.152090i
\(443\) 7.75404i 0.368406i 0.982888 + 0.184203i \(0.0589703\pi\)
−0.982888 + 0.184203i \(0.941030\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −3.66671 −0.173624
\(447\) 0 0
\(448\) − 7.98420i − 0.377218i
\(449\) −33.3401 −1.57342 −0.786709 0.617324i \(-0.788217\pi\)
−0.786709 + 0.617324i \(0.788217\pi\)
\(450\) 0 0
\(451\) 16.8621 0.794004
\(452\) 11.6849i 0.549614i
\(453\) 0 0
\(454\) 5.32898 0.250101
\(455\) 0 0
\(456\) 0 0
\(457\) 38.2192i 1.78782i 0.448246 + 0.893910i \(0.352049\pi\)
−0.448246 + 0.893910i \(0.647951\pi\)
\(458\) 4.95856i 0.231698i
\(459\) 0 0
\(460\) 0 0
\(461\) −31.3033 −1.45794 −0.728971 0.684545i \(-0.760001\pi\)
−0.728971 + 0.684545i \(0.760001\pi\)
\(462\) 0 0
\(463\) 12.0852i 0.561645i 0.959760 + 0.280823i \(0.0906073\pi\)
−0.959760 + 0.280823i \(0.909393\pi\)
\(464\) 4.56352 0.211856
\(465\) 0 0
\(466\) −1.37345 −0.0636238
\(467\) − 7.60466i − 0.351902i −0.984399 0.175951i \(-0.943700\pi\)
0.984399 0.175951i \(-0.0563000\pi\)
\(468\) 0 0
\(469\) −2.51901 −0.116317
\(470\) 0 0
\(471\) 0 0
\(472\) 18.2561i 0.840303i
\(473\) − 47.3636i − 2.17778i
\(474\) 0 0
\(475\) 0 0
\(476\) 14.4237 0.661108
\(477\) 0 0
\(478\) 7.73982i 0.354011i
\(479\) −32.4833 −1.48420 −0.742101 0.670289i \(-0.766170\pi\)
−0.742101 + 0.670289i \(0.766170\pi\)
\(480\) 0 0
\(481\) 7.74608 0.353191
\(482\) 8.29326i 0.377748i
\(483\) 0 0
\(484\) 48.0446 2.18385
\(485\) 0 0
\(486\) 0 0
\(487\) 4.46121i 0.202157i 0.994878 + 0.101078i \(0.0322293\pi\)
−0.994878 + 0.101078i \(0.967771\pi\)
\(488\) − 0.0196451i 0 0.000889291i
\(489\) 0 0
\(490\) 0 0
\(491\) −32.8420 −1.48214 −0.741070 0.671428i \(-0.765681\pi\)
−0.741070 + 0.671428i \(0.765681\pi\)
\(492\) 0 0
\(493\) 5.34300i 0.240637i
\(494\) 0.359515 0.0161754
\(495\) 0 0
\(496\) −22.3501 −1.00355
\(497\) − 16.4934i − 0.739832i
\(498\) 0 0
\(499\) 34.2021 1.53109 0.765547 0.643380i \(-0.222468\pi\)
0.765547 + 0.643380i \(0.222468\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 4.00846i − 0.178906i
\(503\) − 22.1773i − 0.988837i −0.869224 0.494419i \(-0.835381\pi\)
0.869224 0.494419i \(-0.164619\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −12.2990 −0.546758
\(507\) 0 0
\(508\) 16.4432i 0.729549i
\(509\) −21.5632 −0.955773 −0.477887 0.878422i \(-0.658597\pi\)
−0.477887 + 0.878422i \(0.658597\pi\)
\(510\) 0 0
\(511\) 16.9538 0.749990
\(512\) 22.8115i 1.00813i
\(513\) 0 0
\(514\) 1.35943 0.0599617
\(515\) 0 0
\(516\) 0 0
\(517\) − 70.5003i − 3.10060i
\(518\) − 4.40643i − 0.193607i
\(519\) 0 0
\(520\) 0 0
\(521\) −20.2626 −0.887718 −0.443859 0.896097i \(-0.646391\pi\)
−0.443859 + 0.896097i \(0.646391\pi\)
\(522\) 0 0
\(523\) 31.8114i 1.39101i 0.718520 + 0.695507i \(0.244820\pi\)
−0.718520 + 0.695507i \(0.755180\pi\)
\(524\) −0.478457 −0.0209015
\(525\) 0 0
\(526\) 12.1266 0.528745
\(527\) − 26.1676i − 1.13988i
\(528\) 0 0
\(529\) 5.25083 0.228297
\(530\) 0 0
\(531\) 0 0
\(532\) 1.62175i 0.0703117i
\(533\) 5.82870i 0.252469i
\(534\) 0 0
\(535\) 0 0
\(536\) 1.75632 0.0758613
\(537\) 0 0
\(538\) 0.159586i 0.00688024i
\(539\) 2.65717 0.114452
\(540\) 0 0
\(541\) −15.1315 −0.650553 −0.325277 0.945619i \(-0.605457\pi\)
−0.325277 + 0.945619i \(0.605457\pi\)
\(542\) 10.2188i 0.438937i
\(543\) 0 0
\(544\) −15.3831 −0.659543
\(545\) 0 0
\(546\) 0 0
\(547\) 4.08744i 0.174766i 0.996175 + 0.0873831i \(0.0278504\pi\)
−0.996175 + 0.0873831i \(0.972150\pi\)
\(548\) 6.17381i 0.263732i
\(549\) 0 0
\(550\) 0 0
\(551\) −0.600748 −0.0255927
\(552\) 0 0
\(553\) − 24.2757i − 1.03231i
\(554\) 11.4214 0.485249
\(555\) 0 0
\(556\) −26.1926 −1.11082
\(557\) − 13.1425i − 0.556864i −0.960456 0.278432i \(-0.910185\pi\)
0.960456 0.278432i \(-0.0898147\pi\)
\(558\) 0 0
\(559\) 16.3721 0.692467
\(560\) 0 0
\(561\) 0 0
\(562\) − 1.59357i − 0.0672207i
\(563\) 24.5221i 1.03348i 0.856141 + 0.516742i \(0.172855\pi\)
−0.856141 + 0.516742i \(0.827145\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −10.3405 −0.434642
\(567\) 0 0
\(568\) 11.4996i 0.482514i
\(569\) 22.7299 0.952885 0.476442 0.879206i \(-0.341926\pi\)
0.476442 + 0.879206i \(0.341926\pi\)
\(570\) 0 0
\(571\) −0.494186 −0.0206810 −0.0103405 0.999947i \(-0.503292\pi\)
−0.0103405 + 0.999947i \(0.503292\pi\)
\(572\) 23.3607i 0.976758i
\(573\) 0 0
\(574\) 3.31571 0.138395
\(575\) 0 0
\(576\) 0 0
\(577\) 9.41187i 0.391821i 0.980622 + 0.195911i \(0.0627662\pi\)
−0.980622 + 0.195911i \(0.937234\pi\)
\(578\) 3.29382i 0.137005i
\(579\) 0 0
\(580\) 0 0
\(581\) 26.7064 1.10797
\(582\) 0 0
\(583\) 58.2012i 2.41045i
\(584\) −11.8206 −0.489139
\(585\) 0 0
\(586\) 6.50916 0.268891
\(587\) 9.97321i 0.411638i 0.978590 + 0.205819i \(0.0659859\pi\)
−0.978590 + 0.205819i \(0.934014\pi\)
\(588\) 0 0
\(589\) 2.94219 0.121231
\(590\) 0 0
\(591\) 0 0
\(592\) − 9.83144i − 0.404070i
\(593\) − 38.3421i − 1.57452i −0.616621 0.787260i \(-0.711499\pi\)
0.616621 0.787260i \(-0.288501\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −18.0326 −0.738642
\(597\) 0 0
\(598\) − 4.25140i − 0.173852i
\(599\) −10.1533 −0.414852 −0.207426 0.978251i \(-0.566509\pi\)
−0.207426 + 0.978251i \(0.566509\pi\)
\(600\) 0 0
\(601\) −21.2743 −0.867796 −0.433898 0.900962i \(-0.642862\pi\)
−0.433898 + 0.900962i \(0.642862\pi\)
\(602\) − 9.31344i − 0.379587i
\(603\) 0 0
\(604\) −18.3219 −0.745508
\(605\) 0 0
\(606\) 0 0
\(607\) 37.7355i 1.53164i 0.643056 + 0.765819i \(0.277666\pi\)
−0.643056 + 0.765819i \(0.722334\pi\)
\(608\) − 1.72962i − 0.0701452i
\(609\) 0 0
\(610\) 0 0
\(611\) 24.3698 0.985895
\(612\) 0 0
\(613\) − 32.2633i − 1.30310i −0.758605 0.651551i \(-0.774119\pi\)
0.758605 0.651551i \(-0.225881\pi\)
\(614\) 16.1940 0.653536
\(615\) 0 0
\(616\) 28.2537 1.13837
\(617\) − 26.0178i − 1.04744i −0.851891 0.523719i \(-0.824544\pi\)
0.851891 0.523719i \(-0.175456\pi\)
\(618\) 0 0
\(619\) 11.8815 0.477559 0.238780 0.971074i \(-0.423253\pi\)
0.238780 + 0.971074i \(0.423253\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 10.9037i − 0.437197i
\(623\) 16.0688i 0.643783i
\(624\) 0 0
\(625\) 0 0
\(626\) 1.69960 0.0679296
\(627\) 0 0
\(628\) − 1.88723i − 0.0753086i
\(629\) 11.5107 0.458962
\(630\) 0 0
\(631\) −13.2726 −0.528372 −0.264186 0.964472i \(-0.585103\pi\)
−0.264186 + 0.964472i \(0.585103\pi\)
\(632\) 16.9256i 0.673263i
\(633\) 0 0
\(634\) −6.23780 −0.247735
\(635\) 0 0
\(636\) 0 0
\(637\) 0.918501i 0.0363923i
\(638\) 4.92265i 0.194890i
\(639\) 0 0
\(640\) 0 0
\(641\) −44.8149 −1.77008 −0.885042 0.465511i \(-0.845871\pi\)
−0.885042 + 0.465511i \(0.845871\pi\)
\(642\) 0 0
\(643\) − 14.9255i − 0.588605i −0.955712 0.294302i \(-0.904913\pi\)
0.955712 0.294302i \(-0.0950873\pi\)
\(644\) 19.1777 0.755708
\(645\) 0 0
\(646\) 0.534242 0.0210195
\(647\) − 41.2684i − 1.62243i −0.584749 0.811214i \(-0.698807\pi\)
0.584749 0.811214i \(-0.301193\pi\)
\(648\) 0 0
\(649\) 63.0179 2.47367
\(650\) 0 0
\(651\) 0 0
\(652\) − 30.4386i − 1.19207i
\(653\) − 27.6252i − 1.08106i −0.841326 0.540528i \(-0.818224\pi\)
0.841326 0.540528i \(-0.181776\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 7.39788 0.288839
\(657\) 0 0
\(658\) − 13.8630i − 0.540435i
\(659\) 40.0225 1.55905 0.779527 0.626369i \(-0.215460\pi\)
0.779527 + 0.626369i \(0.215460\pi\)
\(660\) 0 0
\(661\) 24.9929 0.972112 0.486056 0.873928i \(-0.338435\pi\)
0.486056 + 0.873928i \(0.338435\pi\)
\(662\) 0.559636i 0.0217509i
\(663\) 0 0
\(664\) −18.6204 −0.722610
\(665\) 0 0
\(666\) 0 0
\(667\) 7.10405i 0.275070i
\(668\) − 7.76738i − 0.300529i
\(669\) 0 0
\(670\) 0 0
\(671\) −0.0678126 −0.00261788
\(672\) 0 0
\(673\) − 40.8048i − 1.57291i −0.617649 0.786454i \(-0.711915\pi\)
0.617649 0.786454i \(-0.288085\pi\)
\(674\) −11.7365 −0.452072
\(675\) 0 0
\(676\) 15.0133 0.577436
\(677\) 41.1894i 1.58304i 0.611146 + 0.791518i \(0.290709\pi\)
−0.611146 + 0.791518i \(0.709291\pi\)
\(678\) 0 0
\(679\) −18.4714 −0.708867
\(680\) 0 0
\(681\) 0 0
\(682\) − 24.1089i − 0.923178i
\(683\) − 1.33820i − 0.0512047i −0.999672 0.0256023i \(-0.991850\pi\)
0.999672 0.0256023i \(-0.00815037\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 9.01333 0.344131
\(687\) 0 0
\(688\) − 20.7798i − 0.792221i
\(689\) −20.1184 −0.766449
\(690\) 0 0
\(691\) −25.2813 −0.961748 −0.480874 0.876790i \(-0.659680\pi\)
−0.480874 + 0.876790i \(0.659680\pi\)
\(692\) 26.0368i 0.989770i
\(693\) 0 0
\(694\) −10.4917 −0.398258
\(695\) 0 0
\(696\) 0 0
\(697\) 8.66148i 0.328077i
\(698\) − 7.06025i − 0.267234i
\(699\) 0 0
\(700\) 0 0
\(701\) 18.2064 0.687645 0.343822 0.939035i \(-0.388278\pi\)
0.343822 + 0.939035i \(0.388278\pi\)
\(702\) 0 0
\(703\) 1.29422i 0.0488126i
\(704\) 19.2159 0.724227
\(705\) 0 0
\(706\) 8.00487 0.301267
\(707\) 17.8669i 0.671953i
\(708\) 0 0
\(709\) −41.8206 −1.57061 −0.785304 0.619111i \(-0.787493\pi\)
−0.785304 + 0.619111i \(0.787493\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 11.2036i − 0.419871i
\(713\) − 34.7925i − 1.30299i
\(714\) 0 0
\(715\) 0 0
\(716\) −12.2016 −0.455995
\(717\) 0 0
\(718\) 0.301301i 0.0112444i
\(719\) 48.9786 1.82660 0.913298 0.407293i \(-0.133527\pi\)
0.913298 + 0.407293i \(0.133527\pi\)
\(720\) 0 0
\(721\) −15.6734 −0.583707
\(722\) − 8.93177i − 0.332406i
\(723\) 0 0
\(724\) −19.4847 −0.724144
\(725\) 0 0
\(726\) 0 0
\(727\) 43.8009i 1.62449i 0.583319 + 0.812243i \(0.301754\pi\)
−0.583319 + 0.812243i \(0.698246\pi\)
\(728\) 9.76642i 0.361968i
\(729\) 0 0
\(730\) 0 0
\(731\) 24.3291 0.899843
\(732\) 0 0
\(733\) − 6.78664i − 0.250670i −0.992114 0.125335i \(-0.959999\pi\)
0.992114 0.125335i \(-0.0400006\pi\)
\(734\) 9.51144 0.351074
\(735\) 0 0
\(736\) −20.4533 −0.753919
\(737\) − 6.06261i − 0.223319i
\(738\) 0 0
\(739\) −28.7245 −1.05665 −0.528324 0.849043i \(-0.677179\pi\)
−0.528324 + 0.849043i \(0.677179\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 11.4445i 0.420142i
\(743\) 31.4523i 1.15387i 0.816789 + 0.576937i \(0.195752\pi\)
−0.816789 + 0.576937i \(0.804248\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 9.29610 0.340354
\(747\) 0 0
\(748\) 34.7141i 1.26927i
\(749\) −37.2537 −1.36122
\(750\) 0 0
\(751\) 10.9532 0.399687 0.199844 0.979828i \(-0.435957\pi\)
0.199844 + 0.979828i \(0.435957\pi\)
\(752\) − 30.9305i − 1.12792i
\(753\) 0 0
\(754\) −1.70161 −0.0619689
\(755\) 0 0
\(756\) 0 0
\(757\) 45.7942i 1.66442i 0.554461 + 0.832210i \(0.312925\pi\)
−0.554461 + 0.832210i \(0.687075\pi\)
\(758\) 3.75868i 0.136521i
\(759\) 0 0
\(760\) 0 0
\(761\) 33.9138 1.22937 0.614687 0.788771i \(-0.289283\pi\)
0.614687 + 0.788771i \(0.289283\pi\)
\(762\) 0 0
\(763\) 4.87528i 0.176497i
\(764\) −24.3894 −0.882378
\(765\) 0 0
\(766\) −14.7602 −0.533309
\(767\) 21.7833i 0.786551i
\(768\) 0 0
\(769\) −7.15972 −0.258186 −0.129093 0.991632i \(-0.541207\pi\)
−0.129093 + 0.991632i \(0.541207\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 0.856710i − 0.0308337i
\(773\) 14.5998i 0.525117i 0.964916 + 0.262558i \(0.0845663\pi\)
−0.964916 + 0.262558i \(0.915434\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 12.8787 0.462318
\(777\) 0 0
\(778\) − 14.8836i − 0.533602i
\(779\) −0.973866 −0.0348924
\(780\) 0 0
\(781\) 39.6954 1.42041
\(782\) − 6.31760i − 0.225917i
\(783\) 0 0
\(784\) 1.16578 0.0416348
\(785\) 0 0
\(786\) 0 0
\(787\) − 18.4728i − 0.658483i −0.944246 0.329242i \(-0.893207\pi\)
0.944246 0.329242i \(-0.106793\pi\)
\(788\) 9.83011i 0.350183i
\(789\) 0 0
\(790\) 0 0
\(791\) 16.8630 0.599578
\(792\) 0 0
\(793\) − 0.0234407i 0 0.000832405i
\(794\) 8.38484 0.297567
\(795\) 0 0
\(796\) −31.0077 −1.09904
\(797\) 41.0374i 1.45362i 0.686838 + 0.726810i \(0.258998\pi\)
−0.686838 + 0.726810i \(0.741002\pi\)
\(798\) 0 0
\(799\) 36.2136 1.28115
\(800\) 0 0
\(801\) 0 0
\(802\) 3.38021i 0.119359i
\(803\) 40.8033i 1.43992i
\(804\) 0 0
\(805\) 0 0
\(806\) 8.33371 0.293543
\(807\) 0 0
\(808\) − 12.4572i − 0.438244i
\(809\) 7.19375 0.252919 0.126459 0.991972i \(-0.459639\pi\)
0.126459 + 0.991972i \(0.459639\pi\)
\(810\) 0 0
\(811\) 38.2183 1.34203 0.671014 0.741445i \(-0.265859\pi\)
0.671014 + 0.741445i \(0.265859\pi\)
\(812\) − 7.67583i − 0.269369i
\(813\) 0 0
\(814\) 10.6051 0.371710
\(815\) 0 0
\(816\) 0 0
\(817\) 2.73547i 0.0957021i
\(818\) − 11.7139i − 0.409566i
\(819\) 0 0
\(820\) 0 0
\(821\) 0.668560 0.0233329 0.0116665 0.999932i \(-0.496286\pi\)
0.0116665 + 0.999932i \(0.496286\pi\)
\(822\) 0 0
\(823\) − 1.42033i − 0.0495096i −0.999694 0.0247548i \(-0.992119\pi\)
0.999694 0.0247548i \(-0.00788051\pi\)
\(824\) 10.9279 0.380690
\(825\) 0 0
\(826\) 12.3917 0.431161
\(827\) 49.8169i 1.73230i 0.499782 + 0.866152i \(0.333414\pi\)
−0.499782 + 0.866152i \(0.666586\pi\)
\(828\) 0 0
\(829\) −36.4150 −1.26475 −0.632373 0.774664i \(-0.717919\pi\)
−0.632373 + 0.774664i \(0.717919\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 6.64235i 0.230282i
\(833\) 1.36490i 0.0472909i
\(834\) 0 0
\(835\) 0 0
\(836\) −3.90312 −0.134992
\(837\) 0 0
\(838\) − 5.04447i − 0.174258i
\(839\) −20.0890 −0.693549 −0.346774 0.937949i \(-0.612723\pi\)
−0.346774 + 0.937949i \(0.612723\pi\)
\(840\) 0 0
\(841\) −26.1566 −0.901953
\(842\) − 3.87056i − 0.133388i
\(843\) 0 0
\(844\) −2.90675 −0.100055
\(845\) 0 0
\(846\) 0 0
\(847\) − 69.3349i − 2.38238i
\(848\) 25.5345i 0.876860i
\(849\) 0 0
\(850\) 0 0
\(851\) 15.3046 0.524637
\(852\) 0 0
\(853\) 26.9084i 0.921326i 0.887575 + 0.460663i \(0.152388\pi\)
−0.887575 + 0.460663i \(0.847612\pi\)
\(854\) −0.0133345 −0.000456296 0
\(855\) 0 0
\(856\) 25.9742 0.887779
\(857\) − 48.8408i − 1.66837i −0.551486 0.834184i \(-0.685939\pi\)
0.551486 0.834184i \(-0.314061\pi\)
\(858\) 0 0
\(859\) 41.4094 1.41287 0.706435 0.707778i \(-0.250302\pi\)
0.706435 + 0.707778i \(0.250302\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 0.791507i − 0.0269588i
\(863\) 50.8101i 1.72960i 0.502119 + 0.864799i \(0.332554\pi\)
−0.502119 + 0.864799i \(0.667446\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 4.71359 0.160174
\(867\) 0 0
\(868\) 37.5928i 1.27598i
\(869\) 58.4252 1.98194
\(870\) 0 0
\(871\) 2.09566 0.0710087
\(872\) − 3.39916i − 0.115110i
\(873\) 0 0
\(874\) 0.710328 0.0240272
\(875\) 0 0
\(876\) 0 0
\(877\) 0.409589i 0.0138308i 0.999976 + 0.00691542i \(0.00220127\pi\)
−0.999976 + 0.00691542i \(0.997799\pi\)
\(878\) − 6.06512i − 0.204688i
\(879\) 0 0
\(880\) 0 0
\(881\) 5.32851 0.179522 0.0897610 0.995963i \(-0.471390\pi\)
0.0897610 + 0.995963i \(0.471390\pi\)
\(882\) 0 0
\(883\) − 14.2064i − 0.478083i −0.971009 0.239042i \(-0.923167\pi\)
0.971009 0.239042i \(-0.0768333\pi\)
\(884\) −11.9996 −0.403590
\(885\) 0 0
\(886\) −3.66964 −0.123284
\(887\) − 7.23193i − 0.242825i −0.992602 0.121412i \(-0.961258\pi\)
0.992602 0.121412i \(-0.0387423\pi\)
\(888\) 0 0
\(889\) 23.7298 0.795871
\(890\) 0 0
\(891\) 0 0
\(892\) 13.7604i 0.460733i
\(893\) 4.07173i 0.136255i
\(894\) 0 0
\(895\) 0 0
\(896\) 28.6650 0.957629
\(897\) 0 0
\(898\) − 15.7784i − 0.526531i
\(899\) −13.9256 −0.464444
\(900\) 0 0
\(901\) −29.8960 −0.995981
\(902\) 7.98006i 0.265707i
\(903\) 0 0
\(904\) −11.7573 −0.391041
\(905\) 0 0
\(906\) 0 0
\(907\) − 38.8101i − 1.28867i −0.764743 0.644335i \(-0.777134\pi\)
0.764743 0.644335i \(-0.222866\pi\)
\(908\) − 19.9986i − 0.663677i
\(909\) 0 0
\(910\) 0 0
\(911\) 40.2781 1.33447 0.667236 0.744846i \(-0.267477\pi\)
0.667236 + 0.744846i \(0.267477\pi\)
\(912\) 0 0
\(913\) 64.2754i 2.12721i
\(914\) −18.0874 −0.598279
\(915\) 0 0
\(916\) 18.6085 0.614842
\(917\) 0.690479i 0.0228016i
\(918\) 0 0
\(919\) 13.0468 0.430375 0.215187 0.976573i \(-0.430964\pi\)
0.215187 + 0.976573i \(0.430964\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 14.8144i − 0.487888i
\(923\) 13.7215i 0.451648i
\(924\) 0 0
\(925\) 0 0
\(926\) −5.71936 −0.187950
\(927\) 0 0
\(928\) 8.18638i 0.268731i
\(929\) 0.293825 0.00964008 0.00482004 0.999988i \(-0.498466\pi\)
0.00482004 + 0.999988i \(0.498466\pi\)
\(930\) 0 0
\(931\) −0.153464 −0.00502959
\(932\) 5.15428i 0.168834i
\(933\) 0 0
\(934\) 3.59894 0.117761
\(935\) 0 0
\(936\) 0 0
\(937\) 16.9141i 0.552559i 0.961077 + 0.276280i \(0.0891016\pi\)
−0.961077 + 0.276280i \(0.910898\pi\)
\(938\) − 1.19213i − 0.0389246i
\(939\) 0 0
\(940\) 0 0
\(941\) −57.2093 −1.86497 −0.932485 0.361209i \(-0.882364\pi\)
−0.932485 + 0.361209i \(0.882364\pi\)
\(942\) 0 0
\(943\) 11.5163i 0.375023i
\(944\) 27.6478 0.899858
\(945\) 0 0
\(946\) 22.4150 0.728775
\(947\) − 38.8746i − 1.26325i −0.775272 0.631627i \(-0.782387\pi\)
0.775272 0.631627i \(-0.217613\pi\)
\(948\) 0 0
\(949\) −14.1044 −0.457850
\(950\) 0 0
\(951\) 0 0
\(952\) 14.5130i 0.470368i
\(953\) 54.4516i 1.76386i 0.471381 + 0.881930i \(0.343756\pi\)
−0.471381 + 0.881930i \(0.656244\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 29.0460 0.939415
\(957\) 0 0
\(958\) − 15.3729i − 0.496675i
\(959\) 8.90965 0.287707
\(960\) 0 0
\(961\) 37.2012 1.20004
\(962\) 3.66587i 0.118192i
\(963\) 0 0
\(964\) 31.1229 1.00240
\(965\) 0 0
\(966\) 0 0
\(967\) 19.5701i 0.629333i 0.949202 + 0.314666i \(0.101893\pi\)
−0.949202 + 0.314666i \(0.898107\pi\)
\(968\) 48.3420i 1.55377i
\(969\) 0 0
\(970\) 0 0
\(971\) −6.31009 −0.202500 −0.101250 0.994861i \(-0.532284\pi\)
−0.101250 + 0.994861i \(0.532284\pi\)
\(972\) 0 0
\(973\) 37.7995i 1.21180i
\(974\) −2.11129 −0.0676500
\(975\) 0 0
\(976\) −0.0297513 −0.000952317 0
\(977\) 7.41911i 0.237358i 0.992933 + 0.118679i \(0.0378660\pi\)
−0.992933 + 0.118679i \(0.962134\pi\)
\(978\) 0 0
\(979\) −38.6734 −1.23601
\(980\) 0 0
\(981\) 0 0
\(982\) − 15.5426i − 0.495986i
\(983\) 10.9827i 0.350295i 0.984542 + 0.175148i \(0.0560403\pi\)
−0.984542 + 0.175148i \(0.943960\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −2.52860 −0.0805270
\(987\) 0 0
\(988\) − 1.34919i − 0.0429234i
\(989\) 32.3479 1.02860
\(990\) 0 0
\(991\) −21.3721 −0.678908 −0.339454 0.940623i \(-0.610242\pi\)
−0.339454 + 0.940623i \(0.610242\pi\)
\(992\) − 40.0932i − 1.27296i
\(993\) 0 0
\(994\) 7.80559 0.247578
\(995\) 0 0
\(996\) 0 0
\(997\) 30.5348i 0.967047i 0.875331 + 0.483524i \(0.160643\pi\)
−0.875331 + 0.483524i \(0.839357\pi\)
\(998\) 16.1863i 0.512368i
\(999\) 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 2025.2.b.o.649.5 8
3.2 odd 2 2025.2.b.n.649.4 8
5.2 odd 4 2025.2.a.z.1.2 4
5.3 odd 4 2025.2.a.p.1.3 4
5.4 even 2 inner 2025.2.b.o.649.4 8
9.2 odd 6 225.2.k.c.49.4 16
9.4 even 3 675.2.k.c.424.4 16
9.5 odd 6 225.2.k.c.124.5 16
9.7 even 3 675.2.k.c.199.5 16
15.2 even 4 2025.2.a.q.1.3 4
15.8 even 4 2025.2.a.y.1.2 4
15.14 odd 2 2025.2.b.n.649.5 8
45.2 even 12 225.2.e.e.76.2 yes 8
45.4 even 6 675.2.k.c.424.5 16
45.7 odd 12 675.2.e.c.226.3 8
45.13 odd 12 675.2.e.e.451.2 8
45.14 odd 6 225.2.k.c.124.4 16
45.22 odd 12 675.2.e.c.451.3 8
45.23 even 12 225.2.e.c.151.3 yes 8
45.29 odd 6 225.2.k.c.49.5 16
45.32 even 12 225.2.e.e.151.2 yes 8
45.34 even 6 675.2.k.c.199.4 16
45.38 even 12 225.2.e.c.76.3 8
45.43 odd 12 675.2.e.e.226.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.e.c.76.3 8 45.38 even 12
225.2.e.c.151.3 yes 8 45.23 even 12
225.2.e.e.76.2 yes 8 45.2 even 12
225.2.e.e.151.2 yes 8 45.32 even 12
225.2.k.c.49.4 16 9.2 odd 6
225.2.k.c.49.5 16 45.29 odd 6
225.2.k.c.124.4 16 45.14 odd 6
225.2.k.c.124.5 16 9.5 odd 6
675.2.e.c.226.3 8 45.7 odd 12
675.2.e.c.451.3 8 45.22 odd 12
675.2.e.e.226.2 8 45.43 odd 12
675.2.e.e.451.2 8 45.13 odd 12
675.2.k.c.199.4 16 45.34 even 6
675.2.k.c.199.5 16 9.7 even 3
675.2.k.c.424.4 16 9.4 even 3
675.2.k.c.424.5 16 45.4 even 6
2025.2.a.p.1.3 4 5.3 odd 4
2025.2.a.q.1.3 4 15.2 even 4
2025.2.a.y.1.2 4 15.8 even 4
2025.2.a.z.1.2 4 5.2 odd 4
2025.2.b.n.649.4 8 3.2 odd 2
2025.2.b.n.649.5 8 15.14 odd 2
2025.2.b.o.649.4 8 5.4 even 2 inner
2025.2.b.o.649.5 8 1.1 even 1 trivial