Properties

Label 1875.2.b.h.1249.8
Level $1875$
Weight $2$
Character 1875.1249
Analytic conductor $14.972$
Analytic rank $0$
Dimension $16$
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] = [1875,2,Mod(1249,1875)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1875, 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("1875.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.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: \(14.9719503790\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 20x^{14} + 156x^{12} + 610x^{10} + 1286x^{8} + 1440x^{6} + 761x^{4} + 130x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.8
Root \(1.08982i\) of defining polynomial
Character \(\chi\) \(=\) 1875.1249
Dual form 1875.2.b.h.1249.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.0898194i q^{2} +1.00000i q^{3} +1.99193 q^{4} +0.0898194 q^{6} +4.36070i q^{7} -0.358553i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-0.0898194i q^{2} +1.00000i q^{3} +1.99193 q^{4} +0.0898194 q^{6} +4.36070i q^{7} -0.358553i q^{8} -1.00000 q^{9} +4.39094 q^{11} +1.99193i q^{12} +1.98166i q^{13} +0.391676 q^{14} +3.95166 q^{16} +0.997022i q^{17} +0.0898194i q^{18} -1.35096 q^{19} -4.36070 q^{21} -0.394392i q^{22} -2.35651i q^{23} +0.358553 q^{24} +0.177991 q^{26} -1.00000i q^{27} +8.68622i q^{28} +7.97856 q^{29} -3.67761 q^{31} -1.07204i q^{32} +4.39094i q^{33} +0.0895519 q^{34} -1.99193 q^{36} -1.43706i q^{37} +0.121342i q^{38} -1.98166 q^{39} -5.98248 q^{41} +0.391676i q^{42} -2.68554i q^{43} +8.74646 q^{44} -0.211660 q^{46} +10.9393i q^{47} +3.95166i q^{48} -12.0157 q^{49} -0.997022 q^{51} +3.94732i q^{52} -11.0510i q^{53} -0.0898194 q^{54} +1.56354 q^{56} -1.35096i q^{57} -0.716629i q^{58} -6.68895 q^{59} -9.45570 q^{61} +0.330321i q^{62} -4.36070i q^{63} +7.80703 q^{64} +0.394392 q^{66} +12.9219i q^{67} +1.98600i q^{68} +2.35651 q^{69} +7.32257 q^{71} +0.358553i q^{72} +0.424804i q^{73} -0.129076 q^{74} -2.69101 q^{76} +19.1476i q^{77} +0.177991i q^{78} +6.35531 q^{79} +1.00000 q^{81} +0.537343i q^{82} -0.737011i q^{83} -8.68622 q^{84} -0.241213 q^{86} +7.97856i q^{87} -1.57439i q^{88} +9.78736 q^{89} -8.64141 q^{91} -4.69401i q^{92} -3.67761i q^{93} +0.982560 q^{94} +1.07204 q^{96} +0.0337081i q^{97} +1.07925i q^{98} -4.39094 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9} + 4 q^{11} - 12 q^{14} + 28 q^{19} - 16 q^{21} + 24 q^{24} + 12 q^{26} - 4 q^{29} - 44 q^{31} + 24 q^{34} + 8 q^{36} + 32 q^{39} + 16 q^{41} - 44 q^{44} - 4 q^{46} - 32 q^{51} + 8 q^{54} + 60 q^{56} - 28 q^{59} - 40 q^{61} - 12 q^{64} - 24 q^{66} + 28 q^{69} + 32 q^{71} - 52 q^{74} - 32 q^{76} + 60 q^{79} + 16 q^{81} + 32 q^{84} + 64 q^{86} - 32 q^{89} - 24 q^{91} - 28 q^{94} + 4 q^{96} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times\).

\(n\) \(626\) \(1252\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.0898194i − 0.0635119i −0.999496 0.0317560i \(-0.989890\pi\)
0.999496 0.0317560i \(-0.0101099\pi\)
\(3\) 1.00000i 0.577350i
\(4\) 1.99193 0.995966
\(5\) 0 0
\(6\) 0.0898194 0.0366686
\(7\) 4.36070i 1.64819i 0.566451 + 0.824095i \(0.308316\pi\)
−0.566451 + 0.824095i \(0.691684\pi\)
\(8\) − 0.358553i − 0.126768i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.39094 1.32392 0.661959 0.749540i \(-0.269725\pi\)
0.661959 + 0.749540i \(0.269725\pi\)
\(12\) 1.99193i 0.575021i
\(13\) 1.98166i 0.549612i 0.961500 + 0.274806i \(0.0886137\pi\)
−0.961500 + 0.274806i \(0.911386\pi\)
\(14\) 0.391676 0.104680
\(15\) 0 0
\(16\) 3.95166 0.987915
\(17\) 0.997022i 0.241813i 0.992664 + 0.120907i \(0.0385802\pi\)
−0.992664 + 0.120907i \(0.961420\pi\)
\(18\) 0.0898194i 0.0211706i
\(19\) −1.35096 −0.309931 −0.154965 0.987920i \(-0.549527\pi\)
−0.154965 + 0.987920i \(0.549527\pi\)
\(20\) 0 0
\(21\) −4.36070 −0.951583
\(22\) − 0.394392i − 0.0840846i
\(23\) − 2.35651i − 0.491366i −0.969350 0.245683i \(-0.920988\pi\)
0.969350 0.245683i \(-0.0790122\pi\)
\(24\) 0.358553 0.0731893
\(25\) 0 0
\(26\) 0.177991 0.0349069
\(27\) − 1.00000i − 0.192450i
\(28\) 8.68622i 1.64154i
\(29\) 7.97856 1.48158 0.740790 0.671736i \(-0.234451\pi\)
0.740790 + 0.671736i \(0.234451\pi\)
\(30\) 0 0
\(31\) −3.67761 −0.660519 −0.330259 0.943890i \(-0.607136\pi\)
−0.330259 + 0.943890i \(0.607136\pi\)
\(32\) − 1.07204i − 0.189512i
\(33\) 4.39094i 0.764365i
\(34\) 0.0895519 0.0153580
\(35\) 0 0
\(36\) −1.99193 −0.331989
\(37\) − 1.43706i − 0.236251i −0.992999 0.118125i \(-0.962312\pi\)
0.992999 0.118125i \(-0.0376885\pi\)
\(38\) 0.121342i 0.0196843i
\(39\) −1.98166 −0.317319
\(40\) 0 0
\(41\) −5.98248 −0.934306 −0.467153 0.884177i \(-0.654720\pi\)
−0.467153 + 0.884177i \(0.654720\pi\)
\(42\) 0.391676i 0.0604369i
\(43\) − 2.68554i − 0.409541i −0.978810 0.204770i \(-0.934355\pi\)
0.978810 0.204770i \(-0.0656447\pi\)
\(44\) 8.74646 1.31858
\(45\) 0 0
\(46\) −0.211660 −0.0312076
\(47\) 10.9393i 1.59566i 0.602883 + 0.797829i \(0.294018\pi\)
−0.602883 + 0.797829i \(0.705982\pi\)
\(48\) 3.95166i 0.570373i
\(49\) −12.0157 −1.71653
\(50\) 0 0
\(51\) −0.997022 −0.139611
\(52\) 3.94732i 0.547395i
\(53\) − 11.0510i − 1.51798i −0.651104 0.758989i \(-0.725694\pi\)
0.651104 0.758989i \(-0.274306\pi\)
\(54\) −0.0898194 −0.0122229
\(55\) 0 0
\(56\) 1.56354 0.208937
\(57\) − 1.35096i − 0.178938i
\(58\) − 0.716629i − 0.0940980i
\(59\) −6.68895 −0.870827 −0.435414 0.900231i \(-0.643398\pi\)
−0.435414 + 0.900231i \(0.643398\pi\)
\(60\) 0 0
\(61\) −9.45570 −1.21068 −0.605339 0.795967i \(-0.706963\pi\)
−0.605339 + 0.795967i \(0.706963\pi\)
\(62\) 0.330321i 0.0419508i
\(63\) − 4.36070i − 0.549397i
\(64\) 7.80703 0.975879
\(65\) 0 0
\(66\) 0.394392 0.0485463
\(67\) 12.9219i 1.57866i 0.613972 + 0.789328i \(0.289571\pi\)
−0.613972 + 0.789328i \(0.710429\pi\)
\(68\) 1.98600i 0.240838i
\(69\) 2.35651 0.283690
\(70\) 0 0
\(71\) 7.32257 0.869029 0.434515 0.900665i \(-0.356920\pi\)
0.434515 + 0.900665i \(0.356920\pi\)
\(72\) 0.358553i 0.0422559i
\(73\) 0.424804i 0.0497195i 0.999691 + 0.0248598i \(0.00791393\pi\)
−0.999691 + 0.0248598i \(0.992086\pi\)
\(74\) −0.129076 −0.0150047
\(75\) 0 0
\(76\) −2.69101 −0.308680
\(77\) 19.1476i 2.18207i
\(78\) 0.177991i 0.0201535i
\(79\) 6.35531 0.715028 0.357514 0.933908i \(-0.383624\pi\)
0.357514 + 0.933908i \(0.383624\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0.537343i 0.0593396i
\(83\) − 0.737011i − 0.0808975i −0.999182 0.0404487i \(-0.987121\pi\)
0.999182 0.0404487i \(-0.0128787\pi\)
\(84\) −8.68622 −0.947745
\(85\) 0 0
\(86\) −0.241213 −0.0260107
\(87\) 7.97856i 0.855391i
\(88\) − 1.57439i − 0.167830i
\(89\) 9.78736 1.03746 0.518729 0.854939i \(-0.326405\pi\)
0.518729 + 0.854939i \(0.326405\pi\)
\(90\) 0 0
\(91\) −8.64141 −0.905866
\(92\) − 4.69401i − 0.489384i
\(93\) − 3.67761i − 0.381351i
\(94\) 0.982560 0.101343
\(95\) 0 0
\(96\) 1.07204 0.109415
\(97\) 0.0337081i 0.00342254i 0.999999 + 0.00171127i \(0.000544714\pi\)
−0.999999 + 0.00171127i \(0.999455\pi\)
\(98\) 1.07925i 0.109020i
\(99\) −4.39094 −0.441306
\(100\) 0 0
\(101\) 3.19390 0.317805 0.158902 0.987294i \(-0.449204\pi\)
0.158902 + 0.987294i \(0.449204\pi\)
\(102\) 0.0895519i 0.00886696i
\(103\) − 8.55342i − 0.842794i −0.906876 0.421397i \(-0.861540\pi\)
0.906876 0.421397i \(-0.138460\pi\)
\(104\) 0.710529 0.0696731
\(105\) 0 0
\(106\) −0.992598 −0.0964097
\(107\) 2.22136i 0.214747i 0.994219 + 0.107373i \(0.0342440\pi\)
−0.994219 + 0.107373i \(0.965756\pi\)
\(108\) − 1.99193i − 0.191674i
\(109\) −11.0023 −1.05383 −0.526916 0.849917i \(-0.676652\pi\)
−0.526916 + 0.849917i \(0.676652\pi\)
\(110\) 0 0
\(111\) 1.43706 0.136399
\(112\) 17.2320i 1.62827i
\(113\) − 1.71021i − 0.160883i −0.996759 0.0804415i \(-0.974367\pi\)
0.996759 0.0804415i \(-0.0256330\pi\)
\(114\) −0.121342 −0.0113647
\(115\) 0 0
\(116\) 15.8927 1.47560
\(117\) − 1.98166i − 0.183204i
\(118\) 0.600798i 0.0553079i
\(119\) −4.34771 −0.398554
\(120\) 0 0
\(121\) 8.28037 0.752761
\(122\) 0.849306i 0.0768925i
\(123\) − 5.98248i − 0.539422i
\(124\) −7.32556 −0.657855
\(125\) 0 0
\(126\) −0.391676 −0.0348933
\(127\) − 12.5570i − 1.11425i −0.830429 0.557125i \(-0.811905\pi\)
0.830429 0.557125i \(-0.188095\pi\)
\(128\) − 2.84531i − 0.251492i
\(129\) 2.68554 0.236448
\(130\) 0 0
\(131\) 16.4718 1.43915 0.719574 0.694416i \(-0.244337\pi\)
0.719574 + 0.694416i \(0.244337\pi\)
\(132\) 8.74646i 0.761282i
\(133\) − 5.89112i − 0.510825i
\(134\) 1.16063 0.100263
\(135\) 0 0
\(136\) 0.357485 0.0306541
\(137\) − 9.66732i − 0.825935i −0.910746 0.412967i \(-0.864492\pi\)
0.910746 0.412967i \(-0.135508\pi\)
\(138\) − 0.211660i − 0.0180177i
\(139\) −13.5327 −1.14783 −0.573913 0.818916i \(-0.694575\pi\)
−0.573913 + 0.818916i \(0.694575\pi\)
\(140\) 0 0
\(141\) −10.9393 −0.921254
\(142\) − 0.657709i − 0.0551937i
\(143\) 8.70133i 0.727642i
\(144\) −3.95166 −0.329305
\(145\) 0 0
\(146\) 0.0381557 0.00315778
\(147\) − 12.0157i − 0.991040i
\(148\) − 2.86252i − 0.235298i
\(149\) −13.6843 −1.12106 −0.560529 0.828134i \(-0.689402\pi\)
−0.560529 + 0.828134i \(0.689402\pi\)
\(150\) 0 0
\(151\) −11.3204 −0.921237 −0.460619 0.887598i \(-0.652372\pi\)
−0.460619 + 0.887598i \(0.652372\pi\)
\(152\) 0.484389i 0.0392892i
\(153\) − 0.997022i − 0.0806044i
\(154\) 1.71983 0.138587
\(155\) 0 0
\(156\) −3.94732 −0.316039
\(157\) 8.56070i 0.683219i 0.939842 + 0.341609i \(0.110972\pi\)
−0.939842 + 0.341609i \(0.889028\pi\)
\(158\) − 0.570830i − 0.0454128i
\(159\) 11.0510 0.876405
\(160\) 0 0
\(161\) 10.2760 0.809865
\(162\) − 0.0898194i − 0.00705688i
\(163\) − 4.58509i − 0.359132i −0.983746 0.179566i \(-0.942531\pi\)
0.983746 0.179566i \(-0.0574694\pi\)
\(164\) −11.9167 −0.930537
\(165\) 0 0
\(166\) −0.0661979 −0.00513795
\(167\) 7.21792i 0.558540i 0.960213 + 0.279270i \(0.0900924\pi\)
−0.960213 + 0.279270i \(0.909908\pi\)
\(168\) 1.56354i 0.120630i
\(169\) 9.07304 0.697926
\(170\) 0 0
\(171\) 1.35096 0.103310
\(172\) − 5.34941i − 0.407889i
\(173\) 4.65009i 0.353540i 0.984252 + 0.176770i \(0.0565649\pi\)
−0.984252 + 0.176770i \(0.943435\pi\)
\(174\) 0.716629 0.0543275
\(175\) 0 0
\(176\) 17.3515 1.30792
\(177\) − 6.68895i − 0.502772i
\(178\) − 0.879095i − 0.0658909i
\(179\) 7.20338 0.538406 0.269203 0.963083i \(-0.413240\pi\)
0.269203 + 0.963083i \(0.413240\pi\)
\(180\) 0 0
\(181\) −3.80424 −0.282767 −0.141383 0.989955i \(-0.545155\pi\)
−0.141383 + 0.989955i \(0.545155\pi\)
\(182\) 0.776167i 0.0575333i
\(183\) − 9.45570i − 0.698986i
\(184\) −0.844934 −0.0622893
\(185\) 0 0
\(186\) −0.330321 −0.0242203
\(187\) 4.37786i 0.320141i
\(188\) 21.7903i 1.58922i
\(189\) 4.36070 0.317194
\(190\) 0 0
\(191\) 21.5541 1.55960 0.779801 0.626027i \(-0.215320\pi\)
0.779801 + 0.626027i \(0.215320\pi\)
\(192\) 7.80703i 0.563424i
\(193\) − 3.15029i − 0.226763i −0.993552 0.113381i \(-0.963832\pi\)
0.993552 0.113381i \(-0.0361682\pi\)
\(194\) 0.00302764 0.000217372 0
\(195\) 0 0
\(196\) −23.9345 −1.70961
\(197\) 26.0837i 1.85839i 0.369590 + 0.929195i \(0.379498\pi\)
−0.369590 + 0.929195i \(0.620502\pi\)
\(198\) 0.394392i 0.0280282i
\(199\) 24.2662 1.72018 0.860092 0.510139i \(-0.170406\pi\)
0.860092 + 0.510139i \(0.170406\pi\)
\(200\) 0 0
\(201\) −12.9219 −0.911437
\(202\) − 0.286874i − 0.0201844i
\(203\) 34.7921i 2.44193i
\(204\) −1.98600 −0.139048
\(205\) 0 0
\(206\) −0.768264 −0.0535275
\(207\) 2.35651i 0.163789i
\(208\) 7.83083i 0.542970i
\(209\) −5.93197 −0.410323
\(210\) 0 0
\(211\) 16.3783 1.12753 0.563765 0.825935i \(-0.309352\pi\)
0.563765 + 0.825935i \(0.309352\pi\)
\(212\) − 22.0129i − 1.51185i
\(213\) 7.32257i 0.501734i
\(214\) 0.199521 0.0136390
\(215\) 0 0
\(216\) −0.358553 −0.0243964
\(217\) − 16.0370i − 1.08866i
\(218\) 0.988224i 0.0669310i
\(219\) −0.424804 −0.0287056
\(220\) 0 0
\(221\) −1.97575 −0.132904
\(222\) − 0.129076i − 0.00866299i
\(223\) 5.68295i 0.380558i 0.981730 + 0.190279i \(0.0609393\pi\)
−0.981730 + 0.190279i \(0.939061\pi\)
\(224\) 4.67486 0.312352
\(225\) 0 0
\(226\) −0.153610 −0.0102180
\(227\) 3.64374i 0.241844i 0.992662 + 0.120922i \(0.0385850\pi\)
−0.992662 + 0.120922i \(0.961415\pi\)
\(228\) − 2.69101i − 0.178217i
\(229\) −1.66125 −0.109779 −0.0548893 0.998492i \(-0.517481\pi\)
−0.0548893 + 0.998492i \(0.517481\pi\)
\(230\) 0 0
\(231\) −19.1476 −1.25982
\(232\) − 2.86074i − 0.187817i
\(233\) − 7.85899i − 0.514860i −0.966297 0.257430i \(-0.917124\pi\)
0.966297 0.257430i \(-0.0828756\pi\)
\(234\) −0.177991 −0.0116356
\(235\) 0 0
\(236\) −13.3239 −0.867314
\(237\) 6.35531i 0.412821i
\(238\) 0.390509i 0.0253130i
\(239\) 0.567301 0.0366956 0.0183478 0.999832i \(-0.494159\pi\)
0.0183478 + 0.999832i \(0.494159\pi\)
\(240\) 0 0
\(241\) −19.0081 −1.22442 −0.612211 0.790695i \(-0.709720\pi\)
−0.612211 + 0.790695i \(0.709720\pi\)
\(242\) − 0.743738i − 0.0478093i
\(243\) 1.00000i 0.0641500i
\(244\) −18.8351 −1.20580
\(245\) 0 0
\(246\) −0.537343 −0.0342597
\(247\) − 2.67713i − 0.170342i
\(248\) 1.31862i 0.0837324i
\(249\) 0.737011 0.0467062
\(250\) 0 0
\(251\) 3.02533 0.190957 0.0954787 0.995431i \(-0.469562\pi\)
0.0954787 + 0.995431i \(0.469562\pi\)
\(252\) − 8.68622i − 0.547181i
\(253\) − 10.3473i − 0.650529i
\(254\) −1.12786 −0.0707681
\(255\) 0 0
\(256\) 15.3585 0.959906
\(257\) − 19.8613i − 1.23891i −0.785032 0.619456i \(-0.787353\pi\)
0.785032 0.619456i \(-0.212647\pi\)
\(258\) − 0.241213i − 0.0150173i
\(259\) 6.26658 0.389386
\(260\) 0 0
\(261\) −7.97856 −0.493860
\(262\) − 1.47949i − 0.0914030i
\(263\) − 22.8299i − 1.40775i −0.710323 0.703876i \(-0.751451\pi\)
0.710323 0.703876i \(-0.248549\pi\)
\(264\) 1.57439 0.0968968
\(265\) 0 0
\(266\) −0.529137 −0.0324435
\(267\) 9.78736i 0.598977i
\(268\) 25.7395i 1.57229i
\(269\) −14.8324 −0.904347 −0.452174 0.891930i \(-0.649351\pi\)
−0.452174 + 0.891930i \(0.649351\pi\)
\(270\) 0 0
\(271\) −6.43720 −0.391032 −0.195516 0.980701i \(-0.562638\pi\)
−0.195516 + 0.980701i \(0.562638\pi\)
\(272\) 3.93989i 0.238891i
\(273\) − 8.64141i − 0.523002i
\(274\) −0.868313 −0.0524567
\(275\) 0 0
\(276\) 4.69401 0.282546
\(277\) 6.70976i 0.403150i 0.979473 + 0.201575i \(0.0646060\pi\)
−0.979473 + 0.201575i \(0.935394\pi\)
\(278\) 1.21550i 0.0729006i
\(279\) 3.67761 0.220173
\(280\) 0 0
\(281\) −20.4867 −1.22214 −0.611068 0.791578i \(-0.709260\pi\)
−0.611068 + 0.791578i \(0.709260\pi\)
\(282\) 0.982560i 0.0585106i
\(283\) 11.4177i 0.678715i 0.940658 + 0.339357i \(0.110210\pi\)
−0.940658 + 0.339357i \(0.889790\pi\)
\(284\) 14.5861 0.865524
\(285\) 0 0
\(286\) 0.781549 0.0462140
\(287\) − 26.0878i − 1.53991i
\(288\) 1.07204i 0.0631707i
\(289\) 16.0059 0.941526
\(290\) 0 0
\(291\) −0.0337081 −0.00197600
\(292\) 0.846181i 0.0495190i
\(293\) − 28.5505i − 1.66794i −0.551812 0.833968i \(-0.686063\pi\)
0.551812 0.833968i \(-0.313937\pi\)
\(294\) −1.07925 −0.0629429
\(295\) 0 0
\(296\) −0.515261 −0.0299489
\(297\) − 4.39094i − 0.254788i
\(298\) 1.22911i 0.0712006i
\(299\) 4.66979 0.270061
\(300\) 0 0
\(301\) 11.7108 0.675001
\(302\) 1.01679i 0.0585095i
\(303\) 3.19390i 0.183485i
\(304\) −5.33852 −0.306185
\(305\) 0 0
\(306\) −0.0895519 −0.00511934
\(307\) − 20.5417i − 1.17238i −0.810175 0.586188i \(-0.800628\pi\)
0.810175 0.586188i \(-0.199372\pi\)
\(308\) 38.1407i 2.17327i
\(309\) 8.55342 0.486587
\(310\) 0 0
\(311\) 17.5496 0.995146 0.497573 0.867422i \(-0.334225\pi\)
0.497573 + 0.867422i \(0.334225\pi\)
\(312\) 0.710529i 0.0402258i
\(313\) 2.98564i 0.168758i 0.996434 + 0.0843790i \(0.0268907\pi\)
−0.996434 + 0.0843790i \(0.973109\pi\)
\(314\) 0.768918 0.0433925
\(315\) 0 0
\(316\) 12.6593 0.712143
\(317\) − 16.1708i − 0.908244i −0.890940 0.454122i \(-0.849953\pi\)
0.890940 0.454122i \(-0.150047\pi\)
\(318\) − 0.992598i − 0.0556621i
\(319\) 35.0334 1.96149
\(320\) 0 0
\(321\) −2.22136 −0.123984
\(322\) − 0.922988i − 0.0514361i
\(323\) − 1.34693i − 0.0749453i
\(324\) 1.99193 0.110663
\(325\) 0 0
\(326\) −0.411830 −0.0228092
\(327\) − 11.0023i − 0.608431i
\(328\) 2.14504i 0.118440i
\(329\) −47.7030 −2.62995
\(330\) 0 0
\(331\) −13.0705 −0.718418 −0.359209 0.933257i \(-0.616953\pi\)
−0.359209 + 0.933257i \(0.616953\pi\)
\(332\) − 1.46808i − 0.0805711i
\(333\) 1.43706i 0.0787502i
\(334\) 0.648310 0.0354739
\(335\) 0 0
\(336\) −17.2320 −0.940083
\(337\) 26.8049i 1.46015i 0.683365 + 0.730077i \(0.260516\pi\)
−0.683365 + 0.730077i \(0.739484\pi\)
\(338\) − 0.814935i − 0.0443266i
\(339\) 1.71021 0.0928858
\(340\) 0 0
\(341\) −16.1482 −0.874473
\(342\) − 0.121342i − 0.00656143i
\(343\) − 21.8721i − 1.18098i
\(344\) −0.962908 −0.0519165
\(345\) 0 0
\(346\) 0.417669 0.0224540
\(347\) − 25.4859i − 1.36815i −0.729410 0.684077i \(-0.760205\pi\)
0.729410 0.684077i \(-0.239795\pi\)
\(348\) 15.8927i 0.851941i
\(349\) 28.0435 1.50113 0.750566 0.660795i \(-0.229781\pi\)
0.750566 + 0.660795i \(0.229781\pi\)
\(350\) 0 0
\(351\) 1.98166 0.105773
\(352\) − 4.70727i − 0.250899i
\(353\) − 14.6667i − 0.780630i −0.920681 0.390315i \(-0.872366\pi\)
0.920681 0.390315i \(-0.127634\pi\)
\(354\) −0.600798 −0.0319320
\(355\) 0 0
\(356\) 19.4958 1.03327
\(357\) − 4.34771i − 0.230105i
\(358\) − 0.647004i − 0.0341952i
\(359\) 14.6205 0.771642 0.385821 0.922574i \(-0.373918\pi\)
0.385821 + 0.922574i \(0.373918\pi\)
\(360\) 0 0
\(361\) −17.1749 −0.903943
\(362\) 0.341694i 0.0179591i
\(363\) 8.28037i 0.434607i
\(364\) −17.2131 −0.902212
\(365\) 0 0
\(366\) −0.849306 −0.0443939
\(367\) − 17.6940i − 0.923617i −0.886980 0.461808i \(-0.847201\pi\)
0.886980 0.461808i \(-0.152799\pi\)
\(368\) − 9.31212i − 0.485428i
\(369\) 5.98248 0.311435
\(370\) 0 0
\(371\) 48.1903 2.50192
\(372\) − 7.32556i − 0.379813i
\(373\) − 12.2293i − 0.633210i −0.948557 0.316605i \(-0.897457\pi\)
0.948557 0.316605i \(-0.102543\pi\)
\(374\) 0.393217 0.0203328
\(375\) 0 0
\(376\) 3.92231 0.202278
\(377\) 15.8107i 0.814295i
\(378\) − 0.391676i − 0.0201456i
\(379\) −28.0951 −1.44315 −0.721574 0.692338i \(-0.756581\pi\)
−0.721574 + 0.692338i \(0.756581\pi\)
\(380\) 0 0
\(381\) 12.5570 0.643312
\(382\) − 1.93598i − 0.0990533i
\(383\) − 34.4334i − 1.75947i −0.475468 0.879733i \(-0.657721\pi\)
0.475468 0.879733i \(-0.342279\pi\)
\(384\) 2.84531 0.145199
\(385\) 0 0
\(386\) −0.282957 −0.0144021
\(387\) 2.68554i 0.136514i
\(388\) 0.0671442i 0.00340873i
\(389\) −13.5225 −0.685619 −0.342809 0.939405i \(-0.611378\pi\)
−0.342809 + 0.939405i \(0.611378\pi\)
\(390\) 0 0
\(391\) 2.34949 0.118819
\(392\) 4.30827i 0.217601i
\(393\) 16.4718i 0.830892i
\(394\) 2.34283 0.118030
\(395\) 0 0
\(396\) −8.74646 −0.439526
\(397\) − 35.9744i − 1.80550i −0.430164 0.902751i \(-0.641544\pi\)
0.430164 0.902751i \(-0.358456\pi\)
\(398\) − 2.17958i − 0.109252i
\(399\) 5.89112 0.294925
\(400\) 0 0
\(401\) 4.35977 0.217717 0.108858 0.994057i \(-0.465281\pi\)
0.108858 + 0.994057i \(0.465281\pi\)
\(402\) 1.16063i 0.0578871i
\(403\) − 7.28776i − 0.363029i
\(404\) 6.36203 0.316523
\(405\) 0 0
\(406\) 3.12501 0.155092
\(407\) − 6.31003i − 0.312777i
\(408\) 0.357485i 0.0176982i
\(409\) 18.1138 0.895668 0.447834 0.894117i \(-0.352196\pi\)
0.447834 + 0.894117i \(0.352196\pi\)
\(410\) 0 0
\(411\) 9.66732 0.476854
\(412\) − 17.0378i − 0.839394i
\(413\) − 29.1685i − 1.43529i
\(414\) 0.211660 0.0104025
\(415\) 0 0
\(416\) 2.12442 0.104158
\(417\) − 13.5327i − 0.662698i
\(418\) 0.532806i 0.0260604i
\(419\) −0.527867 −0.0257880 −0.0128940 0.999917i \(-0.504104\pi\)
−0.0128940 + 0.999917i \(0.504104\pi\)
\(420\) 0 0
\(421\) 18.6586 0.909365 0.454682 0.890654i \(-0.349753\pi\)
0.454682 + 0.890654i \(0.349753\pi\)
\(422\) − 1.47109i − 0.0716117i
\(423\) − 10.9393i − 0.531886i
\(424\) −3.96239 −0.192430
\(425\) 0 0
\(426\) 0.657709 0.0318661
\(427\) − 41.2335i − 1.99543i
\(428\) 4.42480i 0.213881i
\(429\) −8.70133 −0.420104
\(430\) 0 0
\(431\) 20.9913 1.01112 0.505559 0.862792i \(-0.331286\pi\)
0.505559 + 0.862792i \(0.331286\pi\)
\(432\) − 3.95166i − 0.190124i
\(433\) 13.6639i 0.656647i 0.944565 + 0.328324i \(0.106484\pi\)
−0.944565 + 0.328324i \(0.893516\pi\)
\(434\) −1.44043 −0.0691430
\(435\) 0 0
\(436\) −21.9159 −1.04958
\(437\) 3.18354i 0.152289i
\(438\) 0.0381557i 0.00182315i
\(439\) 4.33339 0.206821 0.103411 0.994639i \(-0.467024\pi\)
0.103411 + 0.994639i \(0.467024\pi\)
\(440\) 0 0
\(441\) 12.0157 0.572177
\(442\) 0.177461i 0.00844096i
\(443\) 1.60742i 0.0763707i 0.999271 + 0.0381854i \(0.0121577\pi\)
−0.999271 + 0.0381854i \(0.987842\pi\)
\(444\) 2.86252 0.135849
\(445\) 0 0
\(446\) 0.510439 0.0241700
\(447\) − 13.6843i − 0.647244i
\(448\) 34.0441i 1.60843i
\(449\) 13.8291 0.652634 0.326317 0.945260i \(-0.394192\pi\)
0.326317 + 0.945260i \(0.394192\pi\)
\(450\) 0 0
\(451\) −26.2687 −1.23694
\(452\) − 3.40662i − 0.160234i
\(453\) − 11.3204i − 0.531876i
\(454\) 0.327279 0.0153600
\(455\) 0 0
\(456\) −0.484389 −0.0226836
\(457\) 20.1345i 0.941850i 0.882173 + 0.470925i \(0.156080\pi\)
−0.882173 + 0.470925i \(0.843920\pi\)
\(458\) 0.149213i 0.00697225i
\(459\) 0.997022 0.0465370
\(460\) 0 0
\(461\) 31.5264 1.46833 0.734166 0.678970i \(-0.237573\pi\)
0.734166 + 0.678970i \(0.237573\pi\)
\(462\) 1.71983i 0.0800135i
\(463\) 0.0451697i 0.00209921i 0.999999 + 0.00104961i \(0.000334100\pi\)
−0.999999 + 0.00104961i \(0.999666\pi\)
\(464\) 31.5285 1.46368
\(465\) 0 0
\(466\) −0.705890 −0.0326997
\(467\) − 32.8349i − 1.51942i −0.650265 0.759708i \(-0.725342\pi\)
0.650265 0.759708i \(-0.274658\pi\)
\(468\) − 3.94732i − 0.182465i
\(469\) −56.3484 −2.60193
\(470\) 0 0
\(471\) −8.56070 −0.394456
\(472\) 2.39834i 0.110393i
\(473\) − 11.7920i − 0.542198i
\(474\) 0.570830 0.0262191
\(475\) 0 0
\(476\) −8.66035 −0.396947
\(477\) 11.0510i 0.505992i
\(478\) − 0.0509546i − 0.00233061i
\(479\) −7.48576 −0.342033 −0.171017 0.985268i \(-0.554705\pi\)
−0.171017 + 0.985268i \(0.554705\pi\)
\(480\) 0 0
\(481\) 2.84775 0.129846
\(482\) 1.70730i 0.0777654i
\(483\) 10.2760i 0.467576i
\(484\) 16.4939 0.749724
\(485\) 0 0
\(486\) 0.0898194 0.00407429
\(487\) − 25.4295i − 1.15232i −0.817337 0.576160i \(-0.804551\pi\)
0.817337 0.576160i \(-0.195449\pi\)
\(488\) 3.39037i 0.153475i
\(489\) 4.58509 0.207345
\(490\) 0 0
\(491\) −11.0668 −0.499438 −0.249719 0.968318i \(-0.580338\pi\)
−0.249719 + 0.968318i \(0.580338\pi\)
\(492\) − 11.9167i − 0.537246i
\(493\) 7.95479i 0.358266i
\(494\) −0.240458 −0.0108187
\(495\) 0 0
\(496\) −14.5327 −0.652537
\(497\) 31.9315i 1.43233i
\(498\) − 0.0661979i − 0.00296640i
\(499\) −4.68157 −0.209576 −0.104788 0.994495i \(-0.533416\pi\)
−0.104788 + 0.994495i \(0.533416\pi\)
\(500\) 0 0
\(501\) −7.21792 −0.322473
\(502\) − 0.271734i − 0.0121281i
\(503\) − 10.8285i − 0.482818i −0.970423 0.241409i \(-0.922391\pi\)
0.970423 0.241409i \(-0.0776095\pi\)
\(504\) −1.56354 −0.0696458
\(505\) 0 0
\(506\) −0.929388 −0.0413163
\(507\) 9.07304i 0.402948i
\(508\) − 25.0126i − 1.10975i
\(509\) −11.5007 −0.509757 −0.254879 0.966973i \(-0.582036\pi\)
−0.254879 + 0.966973i \(0.582036\pi\)
\(510\) 0 0
\(511\) −1.85244 −0.0819473
\(512\) − 7.07011i − 0.312457i
\(513\) 1.35096i 0.0596462i
\(514\) −1.78393 −0.0786856
\(515\) 0 0
\(516\) 5.34941 0.235495
\(517\) 48.0338i 2.11252i
\(518\) − 0.562860i − 0.0247307i
\(519\) −4.65009 −0.204116
\(520\) 0 0
\(521\) 0.797807 0.0349525 0.0174763 0.999847i \(-0.494437\pi\)
0.0174763 + 0.999847i \(0.494437\pi\)
\(522\) 0.716629i 0.0313660i
\(523\) − 40.1429i − 1.75533i −0.479279 0.877663i \(-0.659102\pi\)
0.479279 0.877663i \(-0.340898\pi\)
\(524\) 32.8107 1.43334
\(525\) 0 0
\(526\) −2.05057 −0.0894090
\(527\) − 3.66666i − 0.159722i
\(528\) 17.3515i 0.755127i
\(529\) 17.4469 0.758559
\(530\) 0 0
\(531\) 6.68895 0.290276
\(532\) − 11.7347i − 0.508764i
\(533\) − 11.8552i − 0.513506i
\(534\) 0.879095 0.0380422
\(535\) 0 0
\(536\) 4.63317 0.200123
\(537\) 7.20338i 0.310849i
\(538\) 1.33224i 0.0574369i
\(539\) −52.7603 −2.27255
\(540\) 0 0
\(541\) 1.41016 0.0606277 0.0303138 0.999540i \(-0.490349\pi\)
0.0303138 + 0.999540i \(0.490349\pi\)
\(542\) 0.578185i 0.0248352i
\(543\) − 3.80424i − 0.163255i
\(544\) 1.06885 0.0458265
\(545\) 0 0
\(546\) −0.776167 −0.0332169
\(547\) 15.4621i 0.661110i 0.943787 + 0.330555i \(0.107236\pi\)
−0.943787 + 0.330555i \(0.892764\pi\)
\(548\) − 19.2566i − 0.822603i
\(549\) 9.45570 0.403560
\(550\) 0 0
\(551\) −10.7787 −0.459187
\(552\) − 0.844934i − 0.0359628i
\(553\) 27.7136i 1.17850i
\(554\) 0.602667 0.0256049
\(555\) 0 0
\(556\) −26.9562 −1.14320
\(557\) − 18.0445i − 0.764568i −0.924045 0.382284i \(-0.875138\pi\)
0.924045 0.382284i \(-0.124862\pi\)
\(558\) − 0.330321i − 0.0139836i
\(559\) 5.32181 0.225089
\(560\) 0 0
\(561\) −4.37786 −0.184834
\(562\) 1.84011i 0.0776202i
\(563\) 28.5327i 1.20251i 0.799058 + 0.601254i \(0.205332\pi\)
−0.799058 + 0.601254i \(0.794668\pi\)
\(564\) −21.7903 −0.917538
\(565\) 0 0
\(566\) 1.02554 0.0431065
\(567\) 4.36070i 0.183132i
\(568\) − 2.62553i − 0.110165i
\(569\) −16.4072 −0.687826 −0.343913 0.939001i \(-0.611753\pi\)
−0.343913 + 0.939001i \(0.611753\pi\)
\(570\) 0 0
\(571\) −8.34705 −0.349313 −0.174657 0.984629i \(-0.555882\pi\)
−0.174657 + 0.984629i \(0.555882\pi\)
\(572\) 17.3325i 0.724707i
\(573\) 21.5541i 0.900437i
\(574\) −2.34319 −0.0978029
\(575\) 0 0
\(576\) −7.80703 −0.325293
\(577\) 8.43531i 0.351167i 0.984465 + 0.175583i \(0.0561811\pi\)
−0.984465 + 0.175583i \(0.943819\pi\)
\(578\) − 1.43765i − 0.0597982i
\(579\) 3.15029 0.130921
\(580\) 0 0
\(581\) 3.21389 0.133334
\(582\) 0.00302764i 0 0.000125500i
\(583\) − 48.5245i − 2.00968i
\(584\) 0.152315 0.00630283
\(585\) 0 0
\(586\) −2.56439 −0.105934
\(587\) − 24.2852i − 1.00236i −0.865344 0.501179i \(-0.832900\pi\)
0.865344 0.501179i \(-0.167100\pi\)
\(588\) − 23.9345i − 0.987042i
\(589\) 4.96829 0.204715
\(590\) 0 0
\(591\) −26.0837 −1.07294
\(592\) − 5.67876i − 0.233396i
\(593\) 28.4653i 1.16893i 0.811418 + 0.584466i \(0.198696\pi\)
−0.811418 + 0.584466i \(0.801304\pi\)
\(594\) −0.394392 −0.0161821
\(595\) 0 0
\(596\) −27.2581 −1.11654
\(597\) 24.2662i 0.993149i
\(598\) − 0.419438i − 0.0171521i
\(599\) −16.0387 −0.655323 −0.327662 0.944795i \(-0.606261\pi\)
−0.327662 + 0.944795i \(0.606261\pi\)
\(600\) 0 0
\(601\) −8.09005 −0.330000 −0.165000 0.986294i \(-0.552762\pi\)
−0.165000 + 0.986294i \(0.552762\pi\)
\(602\) − 1.05186i − 0.0428706i
\(603\) − 12.9219i − 0.526219i
\(604\) −22.5494 −0.917521
\(605\) 0 0
\(606\) 0.286874 0.0116535
\(607\) − 0.434608i − 0.0176402i −0.999961 0.00882010i \(-0.997192\pi\)
0.999961 0.00882010i \(-0.00280756\pi\)
\(608\) 1.44828i 0.0587356i
\(609\) −34.7921 −1.40985
\(610\) 0 0
\(611\) −21.6779 −0.876994
\(612\) − 1.98600i − 0.0802793i
\(613\) 39.3962i 1.59120i 0.605824 + 0.795599i \(0.292844\pi\)
−0.605824 + 0.795599i \(0.707156\pi\)
\(614\) −1.84504 −0.0744599
\(615\) 0 0
\(616\) 6.86543 0.276616
\(617\) 15.4146i 0.620568i 0.950644 + 0.310284i \(0.100424\pi\)
−0.950644 + 0.310284i \(0.899576\pi\)
\(618\) − 0.768264i − 0.0309041i
\(619\) 10.6718 0.428935 0.214468 0.976731i \(-0.431198\pi\)
0.214468 + 0.976731i \(0.431198\pi\)
\(620\) 0 0
\(621\) −2.35651 −0.0945635
\(622\) − 1.57629i − 0.0632036i
\(623\) 42.6797i 1.70993i
\(624\) −7.83083 −0.313484
\(625\) 0 0
\(626\) 0.268168 0.0107182
\(627\) − 5.93197i − 0.236900i
\(628\) 17.0523i 0.680463i
\(629\) 1.43278 0.0571285
\(630\) 0 0
\(631\) −18.4347 −0.733874 −0.366937 0.930246i \(-0.619594\pi\)
−0.366937 + 0.930246i \(0.619594\pi\)
\(632\) − 2.27871i − 0.0906424i
\(633\) 16.3783i 0.650980i
\(634\) −1.45245 −0.0576843
\(635\) 0 0
\(636\) 22.0129 0.872869
\(637\) − 23.8110i − 0.943427i
\(638\) − 3.14668i − 0.124578i
\(639\) −7.32257 −0.289676
\(640\) 0 0
\(641\) 12.4281 0.490882 0.245441 0.969412i \(-0.421067\pi\)
0.245441 + 0.969412i \(0.421067\pi\)
\(642\) 0.199521i 0.00787447i
\(643\) 1.84657i 0.0728218i 0.999337 + 0.0364109i \(0.0115925\pi\)
−0.999337 + 0.0364109i \(0.988407\pi\)
\(644\) 20.4692 0.806598
\(645\) 0 0
\(646\) −0.120981 −0.00475992
\(647\) − 38.9760i − 1.53230i −0.642660 0.766152i \(-0.722169\pi\)
0.642660 0.766152i \(-0.277831\pi\)
\(648\) − 0.358553i − 0.0140853i
\(649\) −29.3708 −1.15290
\(650\) 0 0
\(651\) 16.0370 0.628539
\(652\) − 9.13319i − 0.357683i
\(653\) − 28.3389i − 1.10899i −0.832189 0.554493i \(-0.812912\pi\)
0.832189 0.554493i \(-0.187088\pi\)
\(654\) −0.988224 −0.0386426
\(655\) 0 0
\(656\) −23.6407 −0.923015
\(657\) − 0.424804i − 0.0165732i
\(658\) 4.28465i 0.167033i
\(659\) 22.3561 0.870868 0.435434 0.900221i \(-0.356595\pi\)
0.435434 + 0.900221i \(0.356595\pi\)
\(660\) 0 0
\(661\) 29.2143 1.13631 0.568153 0.822923i \(-0.307658\pi\)
0.568153 + 0.822923i \(0.307658\pi\)
\(662\) 1.17398i 0.0456281i
\(663\) − 1.97575i − 0.0767319i
\(664\) −0.264258 −0.0102552
\(665\) 0 0
\(666\) 0.129076 0.00500158
\(667\) − 18.8015i − 0.727998i
\(668\) 14.3776i 0.556287i
\(669\) −5.68295 −0.219715
\(670\) 0 0
\(671\) −41.5194 −1.60284
\(672\) 4.67486i 0.180336i
\(673\) − 3.21546i − 0.123947i −0.998078 0.0619735i \(-0.980261\pi\)
0.998078 0.0619735i \(-0.0197394\pi\)
\(674\) 2.40760 0.0927372
\(675\) 0 0
\(676\) 18.0729 0.695111
\(677\) − 47.2470i − 1.81585i −0.419133 0.907925i \(-0.637666\pi\)
0.419133 0.907925i \(-0.362334\pi\)
\(678\) − 0.153610i − 0.00589936i
\(679\) −0.146991 −0.00564099
\(680\) 0 0
\(681\) −3.64374 −0.139629
\(682\) 1.45042i 0.0555395i
\(683\) 37.2527i 1.42543i 0.701452 + 0.712717i \(0.252536\pi\)
−0.701452 + 0.712717i \(0.747464\pi\)
\(684\) 2.69101 0.102893
\(685\) 0 0
\(686\) −1.96454 −0.0750064
\(687\) − 1.66125i − 0.0633807i
\(688\) − 10.6123i − 0.404591i
\(689\) 21.8994 0.834299
\(690\) 0 0
\(691\) −10.9827 −0.417803 −0.208901 0.977937i \(-0.566989\pi\)
−0.208901 + 0.977937i \(0.566989\pi\)
\(692\) 9.26267i 0.352114i
\(693\) − 19.1476i − 0.727357i
\(694\) −2.28913 −0.0868941
\(695\) 0 0
\(696\) 2.86074 0.108436
\(697\) − 5.96466i − 0.225928i
\(698\) − 2.51885i − 0.0953399i
\(699\) 7.85899 0.297254
\(700\) 0 0
\(701\) 22.4086 0.846361 0.423180 0.906046i \(-0.360914\pi\)
0.423180 + 0.906046i \(0.360914\pi\)
\(702\) − 0.177991i − 0.00671784i
\(703\) 1.94140i 0.0732213i
\(704\) 34.2802 1.29198
\(705\) 0 0
\(706\) −1.31736 −0.0495793
\(707\) 13.9276i 0.523803i
\(708\) − 13.3239i − 0.500744i
\(709\) 14.6297 0.549431 0.274716 0.961526i \(-0.411416\pi\)
0.274716 + 0.961526i \(0.411416\pi\)
\(710\) 0 0
\(711\) −6.35531 −0.238343
\(712\) − 3.50929i − 0.131516i
\(713\) 8.66633i 0.324557i
\(714\) −0.390509 −0.0146144
\(715\) 0 0
\(716\) 14.3486 0.536234
\(717\) 0.567301i 0.0211862i
\(718\) − 1.31321i − 0.0490085i
\(719\) 28.4977 1.06278 0.531392 0.847126i \(-0.321669\pi\)
0.531392 + 0.847126i \(0.321669\pi\)
\(720\) 0 0
\(721\) 37.2989 1.38908
\(722\) 1.54264i 0.0574112i
\(723\) − 19.0081i − 0.706920i
\(724\) −7.57778 −0.281626
\(725\) 0 0
\(726\) 0.743738 0.0276027
\(727\) 44.0064i 1.63211i 0.577976 + 0.816054i \(0.303843\pi\)
−0.577976 + 0.816054i \(0.696157\pi\)
\(728\) 3.09840i 0.114835i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 2.67754 0.0990324
\(732\) − 18.8351i − 0.696166i
\(733\) 27.3227i 1.00919i 0.863357 + 0.504593i \(0.168357\pi\)
−0.863357 + 0.504593i \(0.831643\pi\)
\(734\) −1.58926 −0.0586607
\(735\) 0 0
\(736\) −2.52628 −0.0931198
\(737\) 56.7391i 2.09001i
\(738\) − 0.537343i − 0.0197799i
\(739\) −16.4678 −0.605776 −0.302888 0.953026i \(-0.597951\pi\)
−0.302888 + 0.953026i \(0.597951\pi\)
\(740\) 0 0
\(741\) 2.67713 0.0983468
\(742\) − 4.32843i − 0.158901i
\(743\) 35.6012i 1.30608i 0.757322 + 0.653041i \(0.226507\pi\)
−0.757322 + 0.653041i \(0.773493\pi\)
\(744\) −1.31862 −0.0483430
\(745\) 0 0
\(746\) −1.09843 −0.0402164
\(747\) 0.737011i 0.0269658i
\(748\) 8.72041i 0.318850i
\(749\) −9.68668 −0.353944
\(750\) 0 0
\(751\) 46.0748 1.68129 0.840647 0.541583i \(-0.182175\pi\)
0.840647 + 0.541583i \(0.182175\pi\)
\(752\) 43.2283i 1.57638i
\(753\) 3.02533i 0.110249i
\(754\) 1.42011 0.0517174
\(755\) 0 0
\(756\) 8.68622 0.315915
\(757\) 36.6482i 1.33200i 0.745951 + 0.666000i \(0.231995\pi\)
−0.745951 + 0.666000i \(0.768005\pi\)
\(758\) 2.52348i 0.0916571i
\(759\) 10.3473 0.375583
\(760\) 0 0
\(761\) −27.2078 −0.986282 −0.493141 0.869949i \(-0.664151\pi\)
−0.493141 + 0.869949i \(0.664151\pi\)
\(762\) − 1.12786i − 0.0408580i
\(763\) − 47.9779i − 1.73692i
\(764\) 42.9344 1.55331
\(765\) 0 0
\(766\) −3.09279 −0.111747
\(767\) − 13.2552i − 0.478617i
\(768\) 15.3585i 0.554202i
\(769\) −26.5327 −0.956794 −0.478397 0.878144i \(-0.658782\pi\)
−0.478397 + 0.878144i \(0.658782\pi\)
\(770\) 0 0
\(771\) 19.8613 0.715286
\(772\) − 6.27516i − 0.225848i
\(773\) 28.7677i 1.03470i 0.855773 + 0.517351i \(0.173082\pi\)
−0.855773 + 0.517351i \(0.826918\pi\)
\(774\) 0.241213 0.00867024
\(775\) 0 0
\(776\) 0.0120861 0.000433867 0
\(777\) 6.26658i 0.224812i
\(778\) 1.21458i 0.0435450i
\(779\) 8.08206 0.289570
\(780\) 0 0
\(781\) 32.1530 1.15052
\(782\) − 0.211030i − 0.00754641i
\(783\) − 7.97856i − 0.285130i
\(784\) −47.4820 −1.69579
\(785\) 0 0
\(786\) 1.47949 0.0527716
\(787\) 24.8663i 0.886387i 0.896426 + 0.443193i \(0.146155\pi\)
−0.896426 + 0.443193i \(0.853845\pi\)
\(788\) 51.9571i 1.85089i
\(789\) 22.8299 0.812766
\(790\) 0 0
\(791\) 7.45771 0.265166
\(792\) 1.57439i 0.0559434i
\(793\) − 18.7379i − 0.665404i
\(794\) −3.23120 −0.114671
\(795\) 0 0
\(796\) 48.3366 1.71325
\(797\) − 31.4206i − 1.11298i −0.830856 0.556488i \(-0.812149\pi\)
0.830856 0.556488i \(-0.187851\pi\)
\(798\) − 0.529137i − 0.0187312i
\(799\) −10.9067 −0.385851
\(800\) 0 0
\(801\) −9.78736 −0.345819
\(802\) − 0.391592i − 0.0138276i
\(803\) 1.86529i 0.0658246i
\(804\) −25.7395 −0.907761
\(805\) 0 0
\(806\) −0.654583 −0.0230567
\(807\) − 14.8324i − 0.522125i
\(808\) − 1.14518i − 0.0402874i
\(809\) −50.5027 −1.77558 −0.887790 0.460249i \(-0.847760\pi\)
−0.887790 + 0.460249i \(0.847760\pi\)
\(810\) 0 0
\(811\) 25.1612 0.883528 0.441764 0.897131i \(-0.354353\pi\)
0.441764 + 0.897131i \(0.354353\pi\)
\(812\) 69.3035i 2.43208i
\(813\) − 6.43720i − 0.225762i
\(814\) −0.566763 −0.0198650
\(815\) 0 0
\(816\) −3.93989 −0.137924
\(817\) 3.62804i 0.126929i
\(818\) − 1.62697i − 0.0568856i
\(819\) 8.64141 0.301955
\(820\) 0 0
\(821\) −20.6307 −0.720017 −0.360009 0.932949i \(-0.617226\pi\)
−0.360009 + 0.932949i \(0.617226\pi\)
\(822\) − 0.868313i − 0.0302859i
\(823\) 35.8618i 1.25006i 0.780599 + 0.625032i \(0.214914\pi\)
−0.780599 + 0.625032i \(0.785086\pi\)
\(824\) −3.06686 −0.106839
\(825\) 0 0
\(826\) −2.61990 −0.0911580
\(827\) 4.73642i 0.164701i 0.996603 + 0.0823507i \(0.0262428\pi\)
−0.996603 + 0.0823507i \(0.973757\pi\)
\(828\) 4.69401i 0.163128i
\(829\) 28.4768 0.989039 0.494520 0.869166i \(-0.335344\pi\)
0.494520 + 0.869166i \(0.335344\pi\)
\(830\) 0 0
\(831\) −6.70976 −0.232759
\(832\) 15.4708i 0.536355i
\(833\) − 11.9799i − 0.415080i
\(834\) −1.21550 −0.0420892
\(835\) 0 0
\(836\) −11.8161 −0.408668
\(837\) 3.67761i 0.127117i
\(838\) 0.0474127i 0.00163784i
\(839\) 0.765715 0.0264354 0.0132177 0.999913i \(-0.495793\pi\)
0.0132177 + 0.999913i \(0.495793\pi\)
\(840\) 0 0
\(841\) 34.6574 1.19508
\(842\) − 1.67591i − 0.0577555i
\(843\) − 20.4867i − 0.705600i
\(844\) 32.6245 1.12298
\(845\) 0 0
\(846\) −0.982560 −0.0337811
\(847\) 36.1082i 1.24069i
\(848\) − 43.6700i − 1.49963i
\(849\) −11.4177 −0.391856
\(850\) 0 0
\(851\) −3.38644 −0.116086
\(852\) 14.5861i 0.499710i
\(853\) − 34.7289i − 1.18910i −0.804060 0.594548i \(-0.797331\pi\)
0.804060 0.594548i \(-0.202669\pi\)
\(854\) −3.70357 −0.126734
\(855\) 0 0
\(856\) 0.796475 0.0272230
\(857\) 54.2561i 1.85335i 0.375860 + 0.926676i \(0.377347\pi\)
−0.375860 + 0.926676i \(0.622653\pi\)
\(858\) 0.781549i 0.0266816i
\(859\) −15.9095 −0.542824 −0.271412 0.962463i \(-0.587491\pi\)
−0.271412 + 0.962463i \(0.587491\pi\)
\(860\) 0 0
\(861\) 26.0878 0.889070
\(862\) − 1.88543i − 0.0642180i
\(863\) − 22.4714i − 0.764935i −0.923969 0.382467i \(-0.875074\pi\)
0.923969 0.382467i \(-0.124926\pi\)
\(864\) −1.07204 −0.0364716
\(865\) 0 0
\(866\) 1.22729 0.0417049
\(867\) 16.0059i 0.543590i
\(868\) − 31.9446i − 1.08427i
\(869\) 27.9058 0.946639
\(870\) 0 0
\(871\) −25.6067 −0.867649
\(872\) 3.94492i 0.133592i
\(873\) − 0.0337081i − 0.00114085i
\(874\) 0.285944 0.00967219
\(875\) 0 0
\(876\) −0.846181 −0.0285898
\(877\) 27.0756i 0.914278i 0.889395 + 0.457139i \(0.151126\pi\)
−0.889395 + 0.457139i \(0.848874\pi\)
\(878\) − 0.389222i − 0.0131356i
\(879\) 28.5505 0.962984
\(880\) 0 0
\(881\) −7.81317 −0.263232 −0.131616 0.991301i \(-0.542017\pi\)
−0.131616 + 0.991301i \(0.542017\pi\)
\(882\) − 1.07925i − 0.0363401i
\(883\) 58.3010i 1.96199i 0.194042 + 0.980993i \(0.437840\pi\)
−0.194042 + 0.980993i \(0.562160\pi\)
\(884\) −3.93557 −0.132367
\(885\) 0 0
\(886\) 0.144377 0.00485045
\(887\) − 8.64610i − 0.290307i −0.989409 0.145154i \(-0.953632\pi\)
0.989409 0.145154i \(-0.0463677\pi\)
\(888\) − 0.515261i − 0.0172910i
\(889\) 54.7571 1.83650
\(890\) 0 0
\(891\) 4.39094 0.147102
\(892\) 11.3200i 0.379023i
\(893\) − 14.7785i − 0.494543i
\(894\) −1.22911 −0.0411077
\(895\) 0 0
\(896\) 12.4075 0.414507
\(897\) 4.66979i 0.155920i
\(898\) − 1.24212i − 0.0414500i
\(899\) −29.3420 −0.978612
\(900\) 0 0
\(901\) 11.0181 0.367067
\(902\) 2.35944i 0.0785608i
\(903\) 11.7108i 0.389712i
\(904\) −0.613201 −0.0203948
\(905\) 0 0
\(906\) −1.01679 −0.0337805
\(907\) − 40.4367i − 1.34268i −0.741151 0.671339i \(-0.765720\pi\)
0.741151 0.671339i \(-0.234280\pi\)
\(908\) 7.25809i 0.240868i
\(909\) −3.19390 −0.105935
\(910\) 0 0
\(911\) −50.5643 −1.67527 −0.837635 0.546230i \(-0.816063\pi\)
−0.837635 + 0.546230i \(0.816063\pi\)
\(912\) − 5.33852i − 0.176776i
\(913\) − 3.23617i − 0.107102i
\(914\) 1.80847 0.0598187
\(915\) 0 0
\(916\) −3.30910 −0.109336
\(917\) 71.8286i 2.37199i
\(918\) − 0.0895519i − 0.00295565i
\(919\) 43.3566 1.43020 0.715101 0.699021i \(-0.246381\pi\)
0.715101 + 0.699021i \(0.246381\pi\)
\(920\) 0 0
\(921\) 20.5417 0.676871
\(922\) − 2.83169i − 0.0932566i
\(923\) 14.5108i 0.477629i
\(924\) −38.1407 −1.25474
\(925\) 0 0
\(926\) 0.00405712 0.000133325 0
\(927\) 8.55342i 0.280931i
\(928\) − 8.55335i − 0.280777i
\(929\) 34.1746 1.12123 0.560617 0.828075i \(-0.310564\pi\)
0.560617 + 0.828075i \(0.310564\pi\)
\(930\) 0 0
\(931\) 16.2327 0.532006
\(932\) − 15.6546i − 0.512783i
\(933\) 17.5496i 0.574548i
\(934\) −2.94921 −0.0965010
\(935\) 0 0
\(936\) −0.710529 −0.0232244
\(937\) 27.8996i 0.911441i 0.890123 + 0.455720i \(0.150618\pi\)
−0.890123 + 0.455720i \(0.849382\pi\)
\(938\) 5.06118i 0.165253i
\(939\) −2.98564 −0.0974325
\(940\) 0 0
\(941\) −54.3261 −1.77098 −0.885490 0.464659i \(-0.846177\pi\)
−0.885490 + 0.464659i \(0.846177\pi\)
\(942\) 0.768918i 0.0250527i
\(943\) 14.0978i 0.459086i
\(944\) −26.4325 −0.860303
\(945\) 0 0
\(946\) −1.05915 −0.0344361
\(947\) 4.23171i 0.137512i 0.997633 + 0.0687561i \(0.0219030\pi\)
−0.997633 + 0.0687561i \(0.978097\pi\)
\(948\) 12.6593i 0.411156i
\(949\) −0.841815 −0.0273265
\(950\) 0 0
\(951\) 16.1708 0.524375
\(952\) 1.55889i 0.0505238i
\(953\) 47.9307i 1.55263i 0.630346 + 0.776314i \(0.282913\pi\)
−0.630346 + 0.776314i \(0.717087\pi\)
\(954\) 0.992598 0.0321366
\(955\) 0 0
\(956\) 1.13002 0.0365476
\(957\) 35.0334i 1.13247i
\(958\) 0.672367i 0.0217232i
\(959\) 42.1563 1.36130
\(960\) 0 0
\(961\) −17.4752 −0.563715
\(962\) − 0.255783i − 0.00824679i
\(963\) − 2.22136i − 0.0715823i
\(964\) −37.8629 −1.21948
\(965\) 0 0
\(966\) 0.922988 0.0296966
\(967\) − 20.8827i − 0.671544i −0.941943 0.335772i \(-0.891003\pi\)
0.941943 0.335772i \(-0.108997\pi\)
\(968\) − 2.96895i − 0.0954257i
\(969\) 1.34693 0.0432697
\(970\) 0 0
\(971\) 16.0740 0.515838 0.257919 0.966167i \(-0.416963\pi\)
0.257919 + 0.966167i \(0.416963\pi\)
\(972\) 1.99193i 0.0638913i
\(973\) − 59.0119i − 1.89184i
\(974\) −2.28406 −0.0731860
\(975\) 0 0
\(976\) −37.3657 −1.19605
\(977\) − 35.5665i − 1.13787i −0.822381 0.568937i \(-0.807355\pi\)
0.822381 0.568937i \(-0.192645\pi\)
\(978\) − 0.411830i − 0.0131689i
\(979\) 42.9757 1.37351
\(980\) 0 0
\(981\) 11.0023 0.351278
\(982\) 0.994014i 0.0317202i
\(983\) − 18.4711i − 0.589138i −0.955630 0.294569i \(-0.904824\pi\)
0.955630 0.294569i \(-0.0951761\pi\)
\(984\) −2.14504 −0.0683812
\(985\) 0 0
\(986\) 0.714495 0.0227542
\(987\) − 47.7030i − 1.51840i
\(988\) − 5.33266i − 0.169655i
\(989\) −6.32849 −0.201234
\(990\) 0 0
\(991\) −41.5907 −1.32117 −0.660586 0.750750i \(-0.729692\pi\)
−0.660586 + 0.750750i \(0.729692\pi\)
\(992\) 3.94256i 0.125176i
\(993\) − 13.0705i − 0.414779i
\(994\) 2.86807 0.0909698
\(995\) 0 0
\(996\) 1.46808 0.0465178
\(997\) 31.8374i 1.00830i 0.863616 + 0.504151i \(0.168194\pi\)
−0.863616 + 0.504151i \(0.831806\pi\)
\(998\) 0.420496i 0.0133106i
\(999\) −1.43706 −0.0454665
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 1875.2.b.h.1249.8 16
5.2 odd 4 1875.2.a.m.1.5 8
5.3 odd 4 1875.2.a.p.1.4 8
5.4 even 2 inner 1875.2.b.h.1249.9 16
15.2 even 4 5625.2.a.bd.1.4 8
15.8 even 4 5625.2.a.t.1.5 8
25.3 odd 20 375.2.g.d.76.3 16
25.4 even 10 75.2.i.a.34.3 16
25.6 even 5 75.2.i.a.64.3 yes 16
25.8 odd 20 375.2.g.d.301.3 16
25.17 odd 20 375.2.g.e.301.2 16
25.19 even 10 375.2.i.c.199.2 16
25.21 even 5 375.2.i.c.49.2 16
25.22 odd 20 375.2.g.e.76.2 16
75.29 odd 10 225.2.m.b.109.2 16
75.56 odd 10 225.2.m.b.64.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.i.a.34.3 16 25.4 even 10
75.2.i.a.64.3 yes 16 25.6 even 5
225.2.m.b.64.2 16 75.56 odd 10
225.2.m.b.109.2 16 75.29 odd 10
375.2.g.d.76.3 16 25.3 odd 20
375.2.g.d.301.3 16 25.8 odd 20
375.2.g.e.76.2 16 25.22 odd 20
375.2.g.e.301.2 16 25.17 odd 20
375.2.i.c.49.2 16 25.21 even 5
375.2.i.c.199.2 16 25.19 even 10
1875.2.a.m.1.5 8 5.2 odd 4
1875.2.a.p.1.4 8 5.3 odd 4
1875.2.b.h.1249.8 16 1.1 even 1 trivial
1875.2.b.h.1249.9 16 5.4 even 2 inner
5625.2.a.t.1.5 8 15.8 even 4
5625.2.a.bd.1.4 8 15.2 even 4