Properties

Label 1024.2.g.e.129.3
Level $1024$
Weight $2$
Character 1024.129
Analytic conductor $8.177$
Analytic rank $0$
Dimension $16$
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] = [1024,2,Mod(129,1024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1024, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1024.129");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.g (of order \(8\), degree \(4\), 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: \(8.17668116698\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8x^{14} + 32x^{12} - 64x^{10} + 127x^{8} - 576x^{6} + 2592x^{4} - 5832x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 129.3
Root \(-1.50947 + 0.849413i\) of defining polynomial
Character \(\chi\) \(=\) 1024.129
Dual form 1024.2.g.e.897.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.660056 - 1.59352i) q^{3} +(-0.512517 + 0.212292i) q^{5} +(-1.69883 + 1.69883i) q^{7} +(0.0177021 + 0.0177021i) q^{9} +O(q^{10})\) \(q+(0.660056 - 1.59352i) q^{3} +(-0.512517 + 0.212292i) q^{5} +(-1.69883 + 1.69883i) q^{7} +(0.0177021 + 0.0177021i) q^{9} +(1.42542 + 3.44127i) q^{11} +(-5.65154 - 2.34094i) q^{13} +0.956828i q^{15} +5.26768i q^{17} +(3.80083 + 1.57436i) q^{19} +(1.58579 + 3.82843i) q^{21} +(4.31195 + 4.31195i) q^{23} +(-3.31793 + 3.31793i) q^{25} +(4.82044 - 1.99669i) q^{27} +(-0.512517 + 1.23733i) q^{29} -1.53073 q^{31} +6.42458 q^{33} +(0.510031 - 1.23132i) q^{35} +(3.61614 - 1.49785i) q^{37} +(-7.46066 + 7.46066i) q^{39} +(-8.69226 - 8.69226i) q^{41} +(-0.211714 - 0.511123i) q^{43} +(-0.0128306 - 0.00531461i) q^{45} +9.73339i q^{47} +1.22798i q^{49} +(8.39412 + 3.47696i) q^{51} +(2.78771 + 6.73012i) q^{53} +(-1.46111 - 1.46111i) q^{55} +(5.01752 - 5.01752i) q^{57} +(2.76469 - 1.14517i) q^{59} +(3.06575 - 7.40138i) q^{61} -0.0601454 q^{63} +3.39347 q^{65} +(-2.20995 + 5.33529i) q^{67} +(9.71729 - 4.02503i) q^{69} +(9.45315 - 9.45315i) q^{71} +(-1.69977 - 1.69977i) q^{73} +(3.09715 + 7.47719i) q^{75} +(-8.26768 - 3.42458i) q^{77} +12.8332i q^{79} -8.92427i q^{81} +(-0.870678 - 0.360647i) q^{83} +(-1.11828 - 2.69977i) q^{85} +(1.63341 + 1.63341i) q^{87} +(-1.14939 + 1.14939i) q^{89} +(13.5778 - 5.62413i) q^{91} +(-1.01037 + 2.43925i) q^{93} -2.28221 q^{95} -1.83880 q^{97} +(-0.0356847 + 0.0861505i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{5} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{5} - 16 q^{9} - 24 q^{13} + 48 q^{21} - 32 q^{25} + 8 q^{29} + 80 q^{33} + 24 q^{37} - 16 q^{41} - 104 q^{45} + 56 q^{53} - 80 q^{57} - 40 q^{61} + 32 q^{65} - 32 q^{73} - 32 q^{77} + 32 q^{85} + 32 q^{89} - 16 q^{93} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(1023\)
\(\chi(n)\) \(e\left(\frac{7}{8}\right)\) \(1\)

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 0
\(3\) 0.660056 1.59352i 0.381083 0.920017i −0.610673 0.791883i \(-0.709101\pi\)
0.991757 0.128134i \(-0.0408988\pi\)
\(4\) 0 0
\(5\) −0.512517 + 0.212292i −0.229205 + 0.0949397i −0.494330 0.869274i \(-0.664587\pi\)
0.265125 + 0.964214i \(0.414587\pi\)
\(6\) 0 0
\(7\) −1.69883 + 1.69883i −0.642096 + 0.642096i −0.951070 0.308974i \(-0.900014\pi\)
0.308974 + 0.951070i \(0.400014\pi\)
\(8\) 0 0
\(9\) 0.0177021 + 0.0177021i 0.00590068 + 0.00590068i
\(10\) 0 0
\(11\) 1.42542 + 3.44127i 0.429781 + 1.03758i 0.979357 + 0.202139i \(0.0647894\pi\)
−0.549576 + 0.835444i \(0.685211\pi\)
\(12\) 0 0
\(13\) −5.65154 2.34094i −1.56746 0.649261i −0.581091 0.813839i \(-0.697374\pi\)
−0.986364 + 0.164577i \(0.947374\pi\)
\(14\) 0 0
\(15\) 0.956828i 0.247052i
\(16\) 0 0
\(17\) 5.26768i 1.27760i 0.769373 + 0.638799i \(0.220569\pi\)
−0.769373 + 0.638799i \(0.779431\pi\)
\(18\) 0 0
\(19\) 3.80083 + 1.57436i 0.871970 + 0.361182i 0.773377 0.633946i \(-0.218566\pi\)
0.0985928 + 0.995128i \(0.468566\pi\)
\(20\) 0 0
\(21\) 1.58579 + 3.82843i 0.346047 + 0.835431i
\(22\) 0 0
\(23\) 4.31195 + 4.31195i 0.899104 + 0.899104i 0.995357 0.0962527i \(-0.0306857\pi\)
−0.0962527 + 0.995357i \(0.530686\pi\)
\(24\) 0 0
\(25\) −3.31793 + 3.31793i −0.663586 + 0.663586i
\(26\) 0 0
\(27\) 4.82044 1.99669i 0.927694 0.384263i
\(28\) 0 0
\(29\) −0.512517 + 1.23733i −0.0951721 + 0.229766i −0.964295 0.264830i \(-0.914684\pi\)
0.869123 + 0.494596i \(0.164684\pi\)
\(30\) 0 0
\(31\) −1.53073 −0.274928 −0.137464 0.990507i \(-0.543895\pi\)
−0.137464 + 0.990507i \(0.543895\pi\)
\(32\) 0 0
\(33\) 6.42458 1.11838
\(34\) 0 0
\(35\) 0.510031 1.23132i 0.0862110 0.208132i
\(36\) 0 0
\(37\) 3.61614 1.49785i 0.594489 0.246245i −0.0650915 0.997879i \(-0.520734\pi\)
0.659581 + 0.751634i \(0.270734\pi\)
\(38\) 0 0
\(39\) −7.46066 + 7.46066i −1.19466 + 1.19466i
\(40\) 0 0
\(41\) −8.69226 8.69226i −1.35750 1.35750i −0.876985 0.480518i \(-0.840449\pi\)
−0.480518 0.876985i \(-0.659551\pi\)
\(42\) 0 0
\(43\) −0.211714 0.511123i −0.0322861 0.0779456i 0.906914 0.421316i \(-0.138432\pi\)
−0.939200 + 0.343370i \(0.888432\pi\)
\(44\) 0 0
\(45\) −0.0128306 0.00531461i −0.00191267 0.000792255i
\(46\) 0 0
\(47\) 9.73339i 1.41976i 0.704322 + 0.709881i \(0.251251\pi\)
−0.704322 + 0.709881i \(0.748749\pi\)
\(48\) 0 0
\(49\) 1.22798i 0.175425i
\(50\) 0 0
\(51\) 8.39412 + 3.47696i 1.17541 + 0.486872i
\(52\) 0 0
\(53\) 2.78771 + 6.73012i 0.382921 + 0.924454i 0.991398 + 0.130882i \(0.0417809\pi\)
−0.608477 + 0.793572i \(0.708219\pi\)
\(54\) 0 0
\(55\) −1.46111 1.46111i −0.197016 0.197016i
\(56\) 0 0
\(57\) 5.01752 5.01752i 0.664587 0.664587i
\(58\) 0 0
\(59\) 2.76469 1.14517i 0.359933 0.149089i −0.195387 0.980726i \(-0.562596\pi\)
0.555319 + 0.831637i \(0.312596\pi\)
\(60\) 0 0
\(61\) 3.06575 7.40138i 0.392529 0.947650i −0.596858 0.802347i \(-0.703584\pi\)
0.989387 0.145303i \(-0.0464157\pi\)
\(62\) 0 0
\(63\) −0.0601454 −0.00757761
\(64\) 0 0
\(65\) 3.39347 0.420909
\(66\) 0 0
\(67\) −2.20995 + 5.33529i −0.269988 + 0.651810i −0.999482 0.0321778i \(-0.989756\pi\)
0.729494 + 0.683987i \(0.239756\pi\)
\(68\) 0 0
\(69\) 9.71729 4.02503i 1.16982 0.484557i
\(70\) 0 0
\(71\) 9.45315 9.45315i 1.12188 1.12188i 0.130424 0.991458i \(-0.458366\pi\)
0.991458 0.130424i \(-0.0416338\pi\)
\(72\) 0 0
\(73\) −1.69977 1.69977i −0.198944 0.198944i 0.600603 0.799547i \(-0.294927\pi\)
−0.799547 + 0.600603i \(0.794927\pi\)
\(74\) 0 0
\(75\) 3.09715 + 7.47719i 0.357628 + 0.863391i
\(76\) 0 0
\(77\) −8.26768 3.42458i −0.942189 0.390267i
\(78\) 0 0
\(79\) 12.8332i 1.44385i 0.691973 + 0.721923i \(0.256742\pi\)
−0.691973 + 0.721923i \(0.743258\pi\)
\(80\) 0 0
\(81\) 8.92427i 0.991586i
\(82\) 0 0
\(83\) −0.870678 0.360647i −0.0955693 0.0395861i 0.334387 0.942436i \(-0.391471\pi\)
−0.429956 + 0.902850i \(0.641471\pi\)
\(84\) 0 0
\(85\) −1.11828 2.69977i −0.121295 0.292832i
\(86\) 0 0
\(87\) 1.63341 + 1.63341i 0.175120 + 0.175120i
\(88\) 0 0
\(89\) −1.14939 + 1.14939i −0.121835 + 0.121835i −0.765396 0.643560i \(-0.777457\pi\)
0.643560 + 0.765396i \(0.277457\pi\)
\(90\) 0 0
\(91\) 13.5778 5.62413i 1.42334 0.589569i
\(92\) 0 0
\(93\) −1.01037 + 2.43925i −0.104770 + 0.252938i
\(94\) 0 0
\(95\) −2.28221 −0.234150
\(96\) 0 0
\(97\) −1.83880 −0.186702 −0.0933508 0.995633i \(-0.529758\pi\)
−0.0933508 + 0.995633i \(0.529758\pi\)
\(98\) 0 0
\(99\) −0.0356847 + 0.0861505i −0.00358645 + 0.00865845i
\(100\) 0 0
\(101\) −1.76267 + 0.730123i −0.175393 + 0.0726500i −0.468652 0.883383i \(-0.655260\pi\)
0.293259 + 0.956033i \(0.405260\pi\)
\(102\) 0 0
\(103\) −5.36725 + 5.36725i −0.528851 + 0.528851i −0.920230 0.391379i \(-0.871998\pi\)
0.391379 + 0.920230i \(0.371998\pi\)
\(104\) 0 0
\(105\) −1.62549 1.62549i −0.158631 0.158631i
\(106\) 0 0
\(107\) −7.19848 17.3787i −0.695904 1.68006i −0.732534 0.680731i \(-0.761662\pi\)
0.0366299 0.999329i \(-0.488338\pi\)
\(108\) 0 0
\(109\) 6.31591 + 2.61614i 0.604954 + 0.250580i 0.664070 0.747671i \(-0.268828\pi\)
−0.0591151 + 0.998251i \(0.518828\pi\)
\(110\) 0 0
\(111\) 6.75103i 0.640780i
\(112\) 0 0
\(113\) 6.60045i 0.620918i 0.950587 + 0.310459i \(0.100483\pi\)
−0.950587 + 0.310459i \(0.899517\pi\)
\(114\) 0 0
\(115\) −3.12534 1.29456i −0.291440 0.120718i
\(116\) 0 0
\(117\) −0.0586043 0.141483i −0.00541797 0.0130801i
\(118\) 0 0
\(119\) −8.94887 8.94887i −0.820341 0.820341i
\(120\) 0 0
\(121\) −2.03237 + 2.03237i −0.184761 + 0.184761i
\(122\) 0 0
\(123\) −19.5886 + 8.11387i −1.76625 + 0.731603i
\(124\) 0 0
\(125\) 2.05758 4.96745i 0.184036 0.444302i
\(126\) 0 0
\(127\) −4.36789 −0.387588 −0.193794 0.981042i \(-0.562079\pi\)
−0.193794 + 0.981042i \(0.562079\pi\)
\(128\) 0 0
\(129\) −0.954226 −0.0840149
\(130\) 0 0
\(131\) 2.60628 6.29212i 0.227712 0.549745i −0.768186 0.640226i \(-0.778841\pi\)
0.995898 + 0.0904813i \(0.0288405\pi\)
\(132\) 0 0
\(133\) −9.13151 + 3.78239i −0.791802 + 0.327975i
\(134\) 0 0
\(135\) −2.04668 + 2.04668i −0.176150 + 0.176150i
\(136\) 0 0
\(137\) 5.47465 + 5.47465i 0.467731 + 0.467731i 0.901179 0.433448i \(-0.142703\pi\)
−0.433448 + 0.901179i \(0.642703\pi\)
\(138\) 0 0
\(139\) 4.44282 + 10.7259i 0.376835 + 0.909760i 0.992555 + 0.121795i \(0.0388650\pi\)
−0.615721 + 0.787965i \(0.711135\pi\)
\(140\) 0 0
\(141\) 15.5103 + 6.42458i 1.30620 + 0.541048i
\(142\) 0 0
\(143\) 22.7853i 1.90541i
\(144\) 0 0
\(145\) 0.742954i 0.0616990i
\(146\) 0 0
\(147\) 1.95680 + 0.810533i 0.161394 + 0.0668516i
\(148\) 0 0
\(149\) 1.97276 + 4.76267i 0.161615 + 0.390173i 0.983855 0.178968i \(-0.0572758\pi\)
−0.822240 + 0.569141i \(0.807276\pi\)
\(150\) 0 0
\(151\) 13.9603 + 13.9603i 1.13607 + 1.13607i 0.989148 + 0.146924i \(0.0469374\pi\)
0.146924 + 0.989148i \(0.453063\pi\)
\(152\) 0 0
\(153\) −0.0932487 + 0.0932487i −0.00753871 + 0.00753871i
\(154\) 0 0
\(155\) 0.784527 0.324962i 0.0630148 0.0261016i
\(156\) 0 0
\(157\) −6.64117 + 16.0332i −0.530023 + 1.27959i 0.401484 + 0.915866i \(0.368494\pi\)
−0.931507 + 0.363723i \(0.881506\pi\)
\(158\) 0 0
\(159\) 12.5646 0.996438
\(160\) 0 0
\(161\) −14.6505 −1.15462
\(162\) 0 0
\(163\) −1.95307 + 4.71513i −0.152976 + 0.369317i −0.981725 0.190303i \(-0.939053\pi\)
0.828749 + 0.559620i \(0.189053\pi\)
\(164\) 0 0
\(165\) −3.29271 + 1.36388i −0.256337 + 0.106178i
\(166\) 0 0
\(167\) 3.52743 3.52743i 0.272960 0.272960i −0.557330 0.830291i \(-0.688174\pi\)
0.830291 + 0.557330i \(0.188174\pi\)
\(168\) 0 0
\(169\) 17.2675 + 17.2675i 1.32827 + 1.32827i
\(170\) 0 0
\(171\) 0.0394132 + 0.0951518i 0.00301400 + 0.00727644i
\(172\) 0 0
\(173\) −15.1973 6.29491i −1.15543 0.478593i −0.279077 0.960269i \(-0.590028\pi\)
−0.876350 + 0.481676i \(0.840028\pi\)
\(174\) 0 0
\(175\) 11.2732i 0.852171i
\(176\) 0 0
\(177\) 5.16146i 0.387959i
\(178\) 0 0
\(179\) −20.9061 8.65959i −1.56260 0.647248i −0.577057 0.816704i \(-0.695799\pi\)
−0.985538 + 0.169455i \(0.945799\pi\)
\(180\) 0 0
\(181\) −8.60862 20.7830i −0.639874 1.54479i −0.826847 0.562427i \(-0.809868\pi\)
0.186974 0.982365i \(-0.440132\pi\)
\(182\) 0 0
\(183\) −9.77065 9.77065i −0.722267 0.722267i
\(184\) 0 0
\(185\) −1.53535 + 1.53535i −0.112881 + 0.112881i
\(186\) 0 0
\(187\) −18.1275 + 7.50866i −1.32562 + 0.549088i
\(188\) 0 0
\(189\) −4.79706 + 11.5811i −0.348935 + 0.842403i
\(190\) 0 0
\(191\) 10.4366 0.755168 0.377584 0.925975i \(-0.376755\pi\)
0.377584 + 0.925975i \(0.376755\pi\)
\(192\) 0 0
\(193\) −3.81806 −0.274830 −0.137415 0.990514i \(-0.543879\pi\)
−0.137415 + 0.990514i \(0.543879\pi\)
\(194\) 0 0
\(195\) 2.23988 5.40755i 0.160401 0.387243i
\(196\) 0 0
\(197\) −14.6794 + 6.08042i −1.04587 + 0.433212i −0.838414 0.545033i \(-0.816517\pi\)
−0.207452 + 0.978245i \(0.566517\pi\)
\(198\) 0 0
\(199\) 16.0244 16.0244i 1.13594 1.13594i 0.146771 0.989171i \(-0.453112\pi\)
0.989171 0.146771i \(-0.0468879\pi\)
\(200\) 0 0
\(201\) 7.04318 + 7.04318i 0.496788 + 0.496788i
\(202\) 0 0
\(203\) −1.23132 2.97268i −0.0864220 0.208641i
\(204\) 0 0
\(205\) 6.30023 + 2.60964i 0.440027 + 0.182265i
\(206\) 0 0
\(207\) 0.152661i 0.0106107i
\(208\) 0 0
\(209\) 15.3238i 1.05997i
\(210\) 0 0
\(211\) −13.2826 5.50184i −0.914413 0.378762i −0.124669 0.992198i \(-0.539787\pi\)
−0.789744 + 0.613436i \(0.789787\pi\)
\(212\) 0 0
\(213\) −8.82413 21.3033i −0.604620 1.45968i
\(214\) 0 0
\(215\) 0.217014 + 0.217014i 0.0148003 + 0.0148003i
\(216\) 0 0
\(217\) 2.60045 2.60045i 0.176530 0.176530i
\(218\) 0 0
\(219\) −3.83056 + 1.58667i −0.258845 + 0.107217i
\(220\) 0 0
\(221\) 12.3313 29.7705i 0.829495 2.00258i
\(222\) 0 0
\(223\) −16.3670 −1.09602 −0.548008 0.836473i \(-0.684614\pi\)
−0.548008 + 0.836473i \(0.684614\pi\)
\(224\) 0 0
\(225\) −0.117468 −0.00783122
\(226\) 0 0
\(227\) 2.15247 5.19652i 0.142864 0.344905i −0.836210 0.548410i \(-0.815233\pi\)
0.979074 + 0.203505i \(0.0652333\pi\)
\(228\) 0 0
\(229\) −12.1694 + 5.04072i −0.804175 + 0.333100i −0.746627 0.665243i \(-0.768328\pi\)
−0.0575475 + 0.998343i \(0.518328\pi\)
\(230\) 0 0
\(231\) −10.9143 + 10.9143i −0.718105 + 0.718105i
\(232\) 0 0
\(233\) 1.74984 + 1.74984i 0.114636 + 0.114636i 0.762098 0.647462i \(-0.224169\pi\)
−0.647462 + 0.762098i \(0.724169\pi\)
\(234\) 0 0
\(235\) −2.06632 4.98853i −0.134792 0.325416i
\(236\) 0 0
\(237\) 20.4499 + 8.47062i 1.32836 + 0.550226i
\(238\) 0 0
\(239\) 12.7565i 0.825152i −0.910923 0.412576i \(-0.864629\pi\)
0.910923 0.412576i \(-0.135371\pi\)
\(240\) 0 0
\(241\) 18.9660i 1.22171i −0.791743 0.610854i \(-0.790826\pi\)
0.791743 0.610854i \(-0.209174\pi\)
\(242\) 0 0
\(243\) 0.240356 + 0.0995586i 0.0154188 + 0.00638669i
\(244\) 0 0
\(245\) −0.260689 0.629359i −0.0166548 0.0402083i
\(246\) 0 0
\(247\) −17.7951 17.7951i −1.13227 1.13227i
\(248\) 0 0
\(249\) −1.14939 + 1.14939i −0.0728398 + 0.0728398i
\(250\) 0 0
\(251\) −9.10639 + 3.77199i −0.574790 + 0.238086i −0.651092 0.758999i \(-0.725689\pi\)
0.0763018 + 0.997085i \(0.475689\pi\)
\(252\) 0 0
\(253\) −8.69226 + 20.9850i −0.546478 + 1.31931i
\(254\) 0 0
\(255\) −5.04026 −0.315633
\(256\) 0 0
\(257\) −2.06510 −0.128817 −0.0644087 0.997924i \(-0.520516\pi\)
−0.0644087 + 0.997924i \(0.520516\pi\)
\(258\) 0 0
\(259\) −3.59860 + 8.68778i −0.223606 + 0.539832i
\(260\) 0 0
\(261\) −0.0309758 + 0.0128306i −0.00191736 + 0.000794194i
\(262\) 0 0
\(263\) 17.8612 17.8612i 1.10137 1.10137i 0.107120 0.994246i \(-0.465837\pi\)
0.994246 0.107120i \(-0.0341629\pi\)
\(264\) 0 0
\(265\) −2.85750 2.85750i −0.175535 0.175535i
\(266\) 0 0
\(267\) 1.07291 + 2.59024i 0.0656611 + 0.158520i
\(268\) 0 0
\(269\) 13.6544 + 5.65583i 0.832523 + 0.344842i 0.757901 0.652370i \(-0.226225\pi\)
0.0746220 + 0.997212i \(0.476225\pi\)
\(270\) 0 0
\(271\) 2.39655i 0.145580i −0.997347 0.0727901i \(-0.976810\pi\)
0.997347 0.0727901i \(-0.0231903\pi\)
\(272\) 0 0
\(273\) 25.3487i 1.53418i
\(274\) 0 0
\(275\) −16.1473 6.68845i −0.973722 0.403329i
\(276\) 0 0
\(277\) 3.82741 + 9.24018i 0.229967 + 0.555189i 0.996173 0.0874063i \(-0.0278578\pi\)
−0.766206 + 0.642595i \(0.777858\pi\)
\(278\) 0 0
\(279\) −0.0270971 0.0270971i −0.00162226 0.00162226i
\(280\) 0 0
\(281\) 19.8667 19.8667i 1.18515 1.18515i 0.206754 0.978393i \(-0.433710\pi\)
0.978393 0.206754i \(-0.0662900\pi\)
\(282\) 0 0
\(283\) 4.76511 1.97378i 0.283257 0.117329i −0.236532 0.971624i \(-0.576011\pi\)
0.519788 + 0.854295i \(0.326011\pi\)
\(284\) 0 0
\(285\) −1.50639 + 3.63674i −0.0892307 + 0.215422i
\(286\) 0 0
\(287\) 29.5333 1.74329
\(288\) 0 0
\(289\) −10.7484 −0.632259
\(290\) 0 0
\(291\) −1.21371 + 2.93015i −0.0711488 + 0.171769i
\(292\) 0 0
\(293\) 20.2548 8.38983i 1.18330 0.490139i 0.297733 0.954649i \(-0.403770\pi\)
0.885568 + 0.464510i \(0.153770\pi\)
\(294\) 0 0
\(295\) −1.17384 + 1.17384i −0.0683438 + 0.0683438i
\(296\) 0 0
\(297\) 13.7423 + 13.7423i 0.797411 + 0.797411i
\(298\) 0 0
\(299\) −14.2751 34.4632i −0.825552 1.99306i
\(300\) 0 0
\(301\) 1.22798 + 0.508644i 0.0707793 + 0.0293178i
\(302\) 0 0
\(303\) 3.29077i 0.189050i
\(304\) 0 0
\(305\) 4.44417i 0.254472i
\(306\) 0 0
\(307\) 19.9354 + 8.25751i 1.13777 + 0.471281i 0.870416 0.492317i \(-0.163850\pi\)
0.267356 + 0.963598i \(0.413850\pi\)
\(308\) 0 0
\(309\) 5.01011 + 12.0955i 0.285015 + 0.688088i
\(310\) 0 0
\(311\) 15.4798 + 15.4798i 0.877779 + 0.877779i 0.993304 0.115526i \(-0.0368553\pi\)
−0.115526 + 0.993304i \(0.536855\pi\)
\(312\) 0 0
\(313\) −14.0918 + 14.0918i −0.796516 + 0.796516i −0.982544 0.186028i \(-0.940438\pi\)
0.186028 + 0.982544i \(0.440438\pi\)
\(314\) 0 0
\(315\) 0.0308256 0.0127684i 0.00173682 0.000719416i
\(316\) 0 0
\(317\) 3.05683 7.37983i 0.171688 0.414492i −0.814490 0.580177i \(-0.802983\pi\)
0.986179 + 0.165685i \(0.0529834\pi\)
\(318\) 0 0
\(319\) −4.98853 −0.279304
\(320\) 0 0
\(321\) −32.4446 −1.81088
\(322\) 0 0
\(323\) −8.29319 + 20.0215i −0.461445 + 1.11403i
\(324\) 0 0
\(325\) 26.5185 10.9843i 1.47098 0.609300i
\(326\) 0 0
\(327\) 8.33771 8.33771i 0.461076 0.461076i
\(328\) 0 0
\(329\) −16.5354 16.5354i −0.911623 0.911623i
\(330\) 0 0
\(331\) 2.68230 + 6.47564i 0.147433 + 0.355934i 0.980293 0.197550i \(-0.0632983\pi\)
−0.832860 + 0.553483i \(0.813298\pi\)
\(332\) 0 0
\(333\) 0.0905281 + 0.0374980i 0.00496091 + 0.00205488i
\(334\) 0 0
\(335\) 3.20358i 0.175030i
\(336\) 0 0
\(337\) 2.82843i 0.154074i −0.997028 0.0770371i \(-0.975454\pi\)
0.997028 0.0770371i \(-0.0245460\pi\)
\(338\) 0 0
\(339\) 10.5179 + 4.35667i 0.571255 + 0.236622i
\(340\) 0 0
\(341\) −2.18194 5.26768i −0.118159 0.285261i
\(342\) 0 0
\(343\) −13.9779 13.9779i −0.754736 0.754736i
\(344\) 0 0
\(345\) −4.12580 + 4.12580i −0.222126 + 0.222126i
\(346\) 0 0
\(347\) 10.6041 4.39235i 0.569256 0.235794i −0.0794417 0.996840i \(-0.525314\pi\)
0.648698 + 0.761046i \(0.275314\pi\)
\(348\) 0 0
\(349\) 8.18404 19.7580i 0.438081 1.05762i −0.538529 0.842607i \(-0.681020\pi\)
0.976611 0.215015i \(-0.0689801\pi\)
\(350\) 0 0
\(351\) −31.9170 −1.70361
\(352\) 0 0
\(353\) 26.0563 1.38684 0.693418 0.720535i \(-0.256104\pi\)
0.693418 + 0.720535i \(0.256104\pi\)
\(354\) 0 0
\(355\) −2.83808 + 6.85172i −0.150629 + 0.363652i
\(356\) 0 0
\(357\) −20.1669 + 8.35341i −1.06735 + 0.442109i
\(358\) 0 0
\(359\) 6.77460 6.77460i 0.357550 0.357550i −0.505359 0.862909i \(-0.668640\pi\)
0.862909 + 0.505359i \(0.168640\pi\)
\(360\) 0 0
\(361\) −1.46732 1.46732i −0.0772274 0.0772274i
\(362\) 0 0
\(363\) 1.89713 + 4.58008i 0.0995737 + 0.240392i
\(364\) 0 0
\(365\) 1.23201 + 0.510316i 0.0644864 + 0.0267112i
\(366\) 0 0
\(367\) 19.2892i 1.00689i 0.864028 + 0.503444i \(0.167934\pi\)
−0.864028 + 0.503444i \(0.832066\pi\)
\(368\) 0 0
\(369\) 0.307742i 0.0160204i
\(370\) 0 0
\(371\) −16.1691 6.69748i −0.839460 0.347716i
\(372\) 0 0
\(373\) −2.82311 6.81560i −0.146175 0.352898i 0.833786 0.552088i \(-0.186169\pi\)
−0.979961 + 0.199190i \(0.936169\pi\)
\(374\) 0 0
\(375\) −6.55759 6.55759i −0.338632 0.338632i
\(376\) 0 0
\(377\) 5.79302 5.79302i 0.298356 0.298356i
\(378\) 0 0
\(379\) 12.8072 5.30491i 0.657861 0.272495i −0.0286774 0.999589i \(-0.509130\pi\)
0.686538 + 0.727094i \(0.259130\pi\)
\(380\) 0 0
\(381\) −2.88305 + 6.96030i −0.147703 + 0.356587i
\(382\) 0 0
\(383\) 3.78803 0.193559 0.0967797 0.995306i \(-0.469146\pi\)
0.0967797 + 0.995306i \(0.469146\pi\)
\(384\) 0 0
\(385\) 4.96434 0.253006
\(386\) 0 0
\(387\) 0.00530016 0.0127957i 0.000269422 0.000650442i
\(388\) 0 0
\(389\) −19.1472 + 7.93103i −0.970801 + 0.402119i −0.811010 0.585032i \(-0.801082\pi\)
−0.159791 + 0.987151i \(0.551082\pi\)
\(390\) 0 0
\(391\) −22.7140 + 22.7140i −1.14869 + 1.14869i
\(392\) 0 0
\(393\) −8.30630 8.30630i −0.418997 0.418997i
\(394\) 0 0
\(395\) −2.72438 6.57723i −0.137078 0.330936i
\(396\) 0 0
\(397\) 6.41201 + 2.65594i 0.321810 + 0.133298i 0.537740 0.843111i \(-0.319278\pi\)
−0.215930 + 0.976409i \(0.569278\pi\)
\(398\) 0 0
\(399\) 17.0478i 0.853457i
\(400\) 0 0
\(401\) 21.0373i 1.05055i 0.850931 + 0.525277i \(0.176038\pi\)
−0.850931 + 0.525277i \(0.823962\pi\)
\(402\) 0 0
\(403\) 8.65100 + 3.58336i 0.430937 + 0.178500i
\(404\) 0 0
\(405\) 1.89455 + 4.57384i 0.0941408 + 0.227276i
\(406\) 0 0
\(407\) 10.3090 + 10.3090i 0.511000 + 0.511000i
\(408\) 0 0
\(409\) −0.106839 + 0.106839i −0.00528285 + 0.00528285i −0.709743 0.704460i \(-0.751189\pi\)
0.704460 + 0.709743i \(0.251189\pi\)
\(410\) 0 0
\(411\) 12.3375 5.11037i 0.608565 0.252076i
\(412\) 0 0
\(413\) −2.75128 + 6.64219i −0.135382 + 0.326841i
\(414\) 0 0
\(415\) 0.522800 0.0256632
\(416\) 0 0
\(417\) 20.0244 0.980599
\(418\) 0 0
\(419\) 7.75898 18.7318i 0.379051 0.915110i −0.613093 0.790011i \(-0.710075\pi\)
0.992144 0.125099i \(-0.0399250\pi\)
\(420\) 0 0
\(421\) 12.2756 5.08471i 0.598275 0.247814i −0.0629310 0.998018i \(-0.520045\pi\)
0.661206 + 0.750204i \(0.270045\pi\)
\(422\) 0 0
\(423\) −0.172301 + 0.172301i −0.00837756 + 0.00837756i
\(424\) 0 0
\(425\) −17.4778 17.4778i −0.847796 0.847796i
\(426\) 0 0
\(427\) 7.36548 + 17.7819i 0.356441 + 0.860524i
\(428\) 0 0
\(429\) −36.3088 15.0396i −1.75300 0.726118i
\(430\) 0 0
\(431\) 8.96851i 0.431998i 0.976394 + 0.215999i \(0.0693008\pi\)
−0.976394 + 0.215999i \(0.930699\pi\)
\(432\) 0 0
\(433\) 3.14680i 0.151225i −0.997137 0.0756127i \(-0.975909\pi\)
0.997137 0.0756127i \(-0.0240913\pi\)
\(434\) 0 0
\(435\) −1.18391 0.490391i −0.0567641 0.0235125i
\(436\) 0 0
\(437\) 9.60045 + 23.1775i 0.459252 + 1.10873i
\(438\) 0 0
\(439\) 5.35602 + 5.35602i 0.255629 + 0.255629i 0.823274 0.567645i \(-0.192145\pi\)
−0.567645 + 0.823274i \(0.692145\pi\)
\(440\) 0 0
\(441\) −0.0217377 + 0.0217377i −0.00103513 + 0.00103513i
\(442\) 0 0
\(443\) −17.2372 + 7.13988i −0.818964 + 0.339226i −0.752524 0.658564i \(-0.771164\pi\)
−0.0664397 + 0.997790i \(0.521164\pi\)
\(444\) 0 0
\(445\) 0.345077 0.833089i 0.0163582 0.0394922i
\(446\) 0 0
\(447\) 8.89153 0.420555
\(448\) 0 0
\(449\) 35.0511 1.65416 0.827082 0.562081i \(-0.189999\pi\)
0.827082 + 0.562081i \(0.189999\pi\)
\(450\) 0 0
\(451\) 17.5223 42.3026i 0.825093 1.99195i
\(452\) 0 0
\(453\) 31.4605 13.0314i 1.47814 0.612267i
\(454\) 0 0
\(455\) −5.76492 + 5.76492i −0.270264 + 0.270264i
\(456\) 0 0
\(457\) 9.65659 + 9.65659i 0.451716 + 0.451716i 0.895924 0.444207i \(-0.146515\pi\)
−0.444207 + 0.895924i \(0.646515\pi\)
\(458\) 0 0
\(459\) 10.5179 + 25.3925i 0.490935 + 1.18522i
\(460\) 0 0
\(461\) −23.2966 9.64976i −1.08503 0.449434i −0.232759 0.972534i \(-0.574775\pi\)
−0.852271 + 0.523100i \(0.824775\pi\)
\(462\) 0 0
\(463\) 1.57747i 0.0733113i 0.999328 + 0.0366556i \(0.0116705\pi\)
−0.999328 + 0.0366556i \(0.988330\pi\)
\(464\) 0 0
\(465\) 1.46465i 0.0679215i
\(466\) 0 0
\(467\) 23.6421 + 9.79289i 1.09403 + 0.453161i 0.855409 0.517953i \(-0.173306\pi\)
0.238618 + 0.971114i \(0.423306\pi\)
\(468\) 0 0
\(469\) −5.30941 12.8181i −0.245166 0.591883i
\(470\) 0 0
\(471\) 21.1656 + 21.1656i 0.975260 + 0.975260i
\(472\) 0 0
\(473\) 1.45713 1.45713i 0.0669991 0.0669991i
\(474\) 0 0
\(475\) −17.8345 + 7.38728i −0.818302 + 0.338952i
\(476\) 0 0
\(477\) −0.0697888 + 0.168485i −0.00319541 + 0.00771441i
\(478\) 0 0
\(479\) 25.1963 1.15125 0.575623 0.817715i \(-0.304760\pi\)
0.575623 + 0.817715i \(0.304760\pi\)
\(480\) 0 0
\(481\) −23.9431 −1.09171
\(482\) 0 0
\(483\) −9.67016 + 23.3458i −0.440008 + 1.06227i
\(484\) 0 0
\(485\) 0.942415 0.390361i 0.0427929 0.0177254i
\(486\) 0 0
\(487\) −15.4256 + 15.4256i −0.699001 + 0.699001i −0.964195 0.265194i \(-0.914564\pi\)
0.265194 + 0.964195i \(0.414564\pi\)
\(488\) 0 0
\(489\) 6.22450 + 6.22450i 0.281481 + 0.281481i
\(490\) 0 0
\(491\) 11.9376 + 28.8199i 0.538736 + 1.30062i 0.925606 + 0.378489i \(0.123556\pi\)
−0.386870 + 0.922134i \(0.626444\pi\)
\(492\) 0 0
\(493\) −6.51783 2.69977i −0.293548 0.121592i
\(494\) 0 0
\(495\) 0.0517292i 0.00232505i
\(496\) 0 0
\(497\) 32.1185i 1.44071i
\(498\) 0 0
\(499\) 35.2722 + 14.6102i 1.57900 + 0.654043i 0.988255 0.152814i \(-0.0488337\pi\)
0.590746 + 0.806858i \(0.298834\pi\)
\(500\) 0 0
\(501\) −3.29271 7.94930i −0.147107 0.355149i
\(502\) 0 0
\(503\) −2.18240 2.18240i −0.0973084 0.0973084i 0.656777 0.754085i \(-0.271919\pi\)
−0.754085 + 0.656777i \(0.771919\pi\)
\(504\) 0 0
\(505\) 0.748402 0.748402i 0.0333034 0.0333034i
\(506\) 0 0
\(507\) 38.9135 16.1185i 1.72821 0.715848i
\(508\) 0 0
\(509\) 3.08482 7.44742i 0.136732 0.330101i −0.840651 0.541578i \(-0.817827\pi\)
0.977383 + 0.211477i \(0.0678272\pi\)
\(510\) 0 0
\(511\) 5.77524 0.255482
\(512\) 0 0
\(513\) 21.4652 0.947710
\(514\) 0 0
\(515\) 1.61139 3.89023i 0.0710061 0.171424i
\(516\) 0 0
\(517\) −33.4953 + 13.8742i −1.47312 + 0.610187i
\(518\) 0 0
\(519\) −20.0621 + 20.0621i −0.880627 + 0.880627i
\(520\) 0 0
\(521\) −6.94553 6.94553i −0.304289 0.304289i 0.538400 0.842689i \(-0.319029\pi\)
−0.842689 + 0.538400i \(0.819029\pi\)
\(522\) 0 0
\(523\) 0.930625 + 2.24673i 0.0406934 + 0.0982425i 0.942919 0.333021i \(-0.108068\pi\)
−0.902226 + 0.431263i \(0.858068\pi\)
\(524\) 0 0
\(525\) −17.9640 7.44092i −0.784012 0.324748i
\(526\) 0 0
\(527\) 8.06341i 0.351248i
\(528\) 0 0
\(529\) 14.1859i 0.616777i
\(530\) 0 0
\(531\) 0.0692127 + 0.0286688i 0.00300358 + 0.00124412i
\(532\) 0 0
\(533\) 28.7765 + 69.4727i 1.24645 + 3.00920i
\(534\) 0 0
\(535\) 7.37869 + 7.37869i 0.319009 + 0.319009i
\(536\) 0 0
\(537\) −27.5984 + 27.5984i −1.19096 + 1.19096i
\(538\) 0 0
\(539\) −4.22580 + 1.75038i −0.182018 + 0.0753944i
\(540\) 0 0
\(541\) −10.5260 + 25.4120i −0.452548 + 1.09255i 0.518802 + 0.854894i \(0.326378\pi\)
−0.971350 + 0.237653i \(0.923622\pi\)
\(542\) 0 0
\(543\) −38.8003 −1.66508
\(544\) 0 0
\(545\) −3.79240 −0.162448
\(546\) 0 0
\(547\) −1.04627 + 2.52591i −0.0447351 + 0.108000i −0.944668 0.328029i \(-0.893616\pi\)
0.899932 + 0.436029i \(0.143616\pi\)
\(548\) 0 0
\(549\) 0.185290 0.0767495i 0.00790798 0.00327559i
\(550\) 0 0
\(551\) −3.89598 + 3.89598i −0.165974 + 0.165974i
\(552\) 0 0
\(553\) −21.8014 21.8014i −0.927088 0.927088i
\(554\) 0 0
\(555\) 1.43319 + 3.46002i 0.0608354 + 0.146870i
\(556\) 0 0
\(557\) −20.7983 8.61495i −0.881254 0.365027i −0.104271 0.994549i \(-0.533251\pi\)
−0.776983 + 0.629522i \(0.783251\pi\)
\(558\) 0 0
\(559\) 3.38425i 0.143138i
\(560\) 0 0
\(561\) 33.8426i 1.42884i
\(562\) 0 0
\(563\) −20.9200 8.66534i −0.881672 0.365201i −0.104527 0.994522i \(-0.533333\pi\)
−0.777145 + 0.629321i \(0.783333\pi\)
\(564\) 0 0
\(565\) −1.40122 3.38285i −0.0589498 0.142317i
\(566\) 0 0
\(567\) 15.1608 + 15.1608i 0.636693 + 0.636693i
\(568\) 0 0
\(569\) −0.276633 + 0.276633i −0.0115970 + 0.0115970i −0.712882 0.701284i \(-0.752610\pi\)
0.701284 + 0.712882i \(0.252610\pi\)
\(570\) 0 0
\(571\) 31.9788 13.2460i 1.33827 0.554329i 0.405267 0.914198i \(-0.367179\pi\)
0.933003 + 0.359869i \(0.117179\pi\)
\(572\) 0 0
\(573\) 6.88876 16.6309i 0.287782 0.694767i
\(574\) 0 0
\(575\) −28.6135 −1.19327
\(576\) 0 0
\(577\) 30.0407 1.25061 0.625306 0.780379i \(-0.284974\pi\)
0.625306 + 0.780379i \(0.284974\pi\)
\(578\) 0 0
\(579\) −2.52013 + 6.08413i −0.104733 + 0.252848i
\(580\) 0 0
\(581\) 2.09181 0.866455i 0.0867828 0.0359466i
\(582\) 0 0
\(583\) −19.1865 + 19.1865i −0.794625 + 0.794625i
\(584\) 0 0
\(585\) 0.0600715 + 0.0600715i 0.00248365 + 0.00248365i
\(586\) 0 0
\(587\) −13.5838 32.7941i −0.560662 1.35356i −0.909238 0.416277i \(-0.863335\pi\)
0.348575 0.937281i \(-0.386665\pi\)
\(588\) 0 0
\(589\) −5.81806 2.40992i −0.239729 0.0992990i
\(590\) 0 0
\(591\) 27.4053i 1.12730i
\(592\) 0 0
\(593\) 2.66555i 0.109461i 0.998501 + 0.0547306i \(0.0174300\pi\)
−0.998501 + 0.0547306i \(0.982570\pi\)
\(594\) 0 0
\(595\) 6.48622 + 2.68668i 0.265909 + 0.110143i
\(596\) 0 0
\(597\) −14.9582 36.1122i −0.612197 1.47797i
\(598\) 0 0
\(599\) −11.9077 11.9077i −0.486534 0.486534i 0.420677 0.907211i \(-0.361793\pi\)
−0.907211 + 0.420677i \(0.861793\pi\)
\(600\) 0 0
\(601\) −1.86501 + 1.86501i −0.0760755 + 0.0760755i −0.744121 0.668045i \(-0.767131\pi\)
0.668045 + 0.744121i \(0.267131\pi\)
\(602\) 0 0
\(603\) −0.133566 + 0.0553250i −0.00543924 + 0.00225301i
\(604\) 0 0
\(605\) 0.610169 1.47308i 0.0248069 0.0598891i
\(606\) 0 0
\(607\) 8.47275 0.343898 0.171949 0.985106i \(-0.444994\pi\)
0.171949 + 0.985106i \(0.444994\pi\)
\(608\) 0 0
\(609\) −5.54976 −0.224887
\(610\) 0 0
\(611\) 22.7853 55.0087i 0.921796 2.22541i
\(612\) 0 0
\(613\) 23.3613 9.67657i 0.943555 0.390833i 0.142750 0.989759i \(-0.454405\pi\)
0.800805 + 0.598926i \(0.204405\pi\)
\(614\) 0 0
\(615\) 8.31700 8.31700i 0.335374 0.335374i
\(616\) 0 0
\(617\) −8.91738 8.91738i −0.359000 0.359000i 0.504444 0.863444i \(-0.331697\pi\)
−0.863444 + 0.504444i \(0.831697\pi\)
\(618\) 0 0
\(619\) 2.30623 + 5.56774i 0.0926953 + 0.223786i 0.963426 0.267973i \(-0.0863539\pi\)
−0.870731 + 0.491760i \(0.836354\pi\)
\(620\) 0 0
\(621\) 29.3951 + 12.1759i 1.17959 + 0.488601i
\(622\) 0 0
\(623\) 3.90524i 0.156460i
\(624\) 0 0
\(625\) 20.4786i 0.819143i
\(626\) 0 0
\(627\) 24.4187 + 10.1146i 0.975191 + 0.403937i
\(628\) 0 0
\(629\) 7.89020 + 19.0486i 0.314603 + 0.759519i
\(630\) 0 0
\(631\) 33.4932 + 33.4932i 1.33334 + 1.33334i 0.902361 + 0.430981i \(0.141833\pi\)
0.430981 + 0.902361i \(0.358167\pi\)
\(632\) 0 0
\(633\) −17.5345 + 17.5345i −0.696935 + 0.696935i
\(634\) 0 0
\(635\) 2.23862 0.927266i 0.0888369 0.0367974i
\(636\) 0 0
\(637\) 2.87462 6.93995i 0.113897 0.274971i
\(638\) 0 0
\(639\) 0.334680 0.0132397
\(640\) 0 0
\(641\) −26.8508 −1.06054 −0.530272 0.847827i \(-0.677910\pi\)
−0.530272 + 0.847827i \(0.677910\pi\)
\(642\) 0 0
\(643\) −13.0339 + 31.4666i −0.514006 + 1.24092i 0.427528 + 0.904002i \(0.359384\pi\)
−0.941534 + 0.336918i \(0.890616\pi\)
\(644\) 0 0
\(645\) 0.489057 0.202574i 0.0192566 0.00797635i
\(646\) 0 0
\(647\) 6.43000 6.43000i 0.252789 0.252789i −0.569324 0.822113i \(-0.692795\pi\)
0.822113 + 0.569324i \(0.192795\pi\)
\(648\) 0 0
\(649\) 7.88172 + 7.88172i 0.309384 + 0.309384i
\(650\) 0 0
\(651\) −2.42742 5.86030i −0.0951380 0.229683i
\(652\) 0 0
\(653\) −12.6294 5.23125i −0.494225 0.204715i 0.121628 0.992576i \(-0.461188\pi\)
−0.615853 + 0.787861i \(0.711188\pi\)
\(654\) 0 0
\(655\) 3.77811i 0.147623i
\(656\) 0 0
\(657\) 0.0601790i 0.00234781i
\(658\) 0 0
\(659\) 14.3485 + 5.94335i 0.558939 + 0.231520i 0.644224 0.764837i \(-0.277180\pi\)
−0.0852857 + 0.996357i \(0.527180\pi\)
\(660\) 0 0
\(661\) 8.79664 + 21.2370i 0.342150 + 0.826022i 0.997498 + 0.0706953i \(0.0225218\pi\)
−0.655348 + 0.755327i \(0.727478\pi\)
\(662\) 0 0
\(663\) −39.3003 39.3003i −1.52630 1.52630i
\(664\) 0 0
\(665\) 3.87708 3.87708i 0.150347 0.150347i
\(666\) 0 0
\(667\) −7.54524 + 3.12534i −0.292153 + 0.121014i
\(668\) 0 0
\(669\) −10.8031 + 26.0811i −0.417673 + 1.00835i
\(670\) 0 0
\(671\) 29.8402 1.15197
\(672\) 0 0
\(673\) 39.6959 1.53017 0.765083 0.643932i \(-0.222698\pi\)
0.765083 + 0.643932i \(0.222698\pi\)
\(674\) 0 0
\(675\) −9.36899 + 22.6187i −0.360613 + 0.870596i
\(676\) 0 0
\(677\) 19.7220 8.16911i 0.757977 0.313964i 0.0299855 0.999550i \(-0.490454\pi\)
0.727992 + 0.685586i \(0.240454\pi\)
\(678\) 0 0
\(679\) 3.12380 3.12380i 0.119880 0.119880i
\(680\) 0 0
\(681\) −6.85998 6.85998i −0.262875 0.262875i
\(682\) 0 0
\(683\) 9.39365 + 22.6783i 0.359438 + 0.867760i 0.995379 + 0.0960227i \(0.0306121\pi\)
−0.635941 + 0.771738i \(0.719388\pi\)
\(684\) 0 0
\(685\) −3.96808 1.64363i −0.151612 0.0627999i
\(686\) 0 0
\(687\) 22.7192i 0.866793i
\(688\) 0 0
\(689\) 44.5614i 1.69766i
\(690\) 0 0
\(691\) 13.6135 + 5.63891i 0.517883 + 0.214514i 0.626287 0.779593i \(-0.284574\pi\)
−0.108404 + 0.994107i \(0.534574\pi\)
\(692\) 0 0
\(693\) −0.0857327 0.206977i −0.00325671 0.00786241i
\(694\) 0 0
\(695\) −4.55404 4.55404i −0.172745 0.172745i
\(696\) 0 0
\(697\) 45.7880 45.7880i 1.73434 1.73434i
\(698\) 0 0
\(699\) 3.94340 1.63341i 0.149153 0.0617812i
\(700\) 0 0
\(701\) 17.6385 42.5832i 0.666199 1.60835i −0.121718 0.992565i \(-0.538840\pi\)
0.787917 0.615781i \(-0.211160\pi\)
\(702\) 0 0
\(703\) 16.1025 0.607316
\(704\) 0 0
\(705\) −9.31319 −0.350755
\(706\) 0 0
\(707\) 1.75412 4.23483i 0.0659706 0.159267i
\(708\) 0 0
\(709\) 26.8688 11.1294i 1.00908 0.417974i 0.183960 0.982934i \(-0.441108\pi\)
0.825118 + 0.564960i \(0.191108\pi\)
\(710\) 0 0
\(711\) −0.227174 + 0.227174i −0.00851968 + 0.00851968i
\(712\) 0 0
\(713\) −6.60045 6.60045i −0.247189 0.247189i
\(714\) 0 0
\(715\) 4.83714 + 11.6779i 0.180899 + 0.436728i
\(716\) 0 0
\(717\) −20.3277 8.42003i −0.759154 0.314452i
\(718\) 0 0
\(719\) 1.82423i 0.0680323i −0.999421 0.0340162i \(-0.989170\pi\)
0.999421 0.0340162i \(-0.0108298\pi\)
\(720\) 0 0
\(721\) 18.2360i 0.679146i
\(722\) 0 0
\(723\) −30.2226 12.5186i −1.12399 0.465573i
\(724\) 0 0
\(725\) −2.40486 5.80585i −0.0893144 0.215624i
\(726\) 0 0
\(727\) −20.1795 20.1795i −0.748416 0.748416i 0.225765 0.974182i \(-0.427512\pi\)
−0.974182 + 0.225765i \(0.927512\pi\)
\(728\) 0 0
\(729\) 19.2485 19.2485i 0.712909 0.712909i
\(730\) 0 0
\(731\) 2.69243 1.11524i 0.0995832 0.0412487i
\(732\) 0 0
\(733\) 4.66073 11.2520i 0.172148 0.415602i −0.814133 0.580679i \(-0.802787\pi\)
0.986281 + 0.165077i \(0.0527873\pi\)
\(734\) 0 0
\(735\) −1.17496 −0.0433391
\(736\) 0 0
\(737\) −21.5103 −0.792343
\(738\) 0 0
\(739\) −9.38552 + 22.6586i −0.345252 + 0.833512i 0.651915 + 0.758292i \(0.273966\pi\)
−0.997167 + 0.0752197i \(0.976034\pi\)
\(740\) 0 0
\(741\) −40.1024 + 16.6110i −1.47320 + 0.610219i
\(742\) 0 0
\(743\) 30.4077 30.4077i 1.11555 1.11555i 0.123164 0.992386i \(-0.460696\pi\)
0.992386 0.123164i \(-0.0393041\pi\)
\(744\) 0 0
\(745\) −2.02215 2.02215i −0.0740859 0.0740859i
\(746\) 0 0
\(747\) −0.00902860 0.0217970i −0.000330339 0.000797510i
\(748\) 0 0
\(749\) 41.7523 + 17.2944i 1.52560 + 0.631923i
\(750\) 0 0
\(751\) 5.81357i 0.212140i 0.994359 + 0.106070i \(0.0338268\pi\)
−0.994359 + 0.106070i \(0.966173\pi\)
\(752\) 0 0
\(753\) 17.0009i 0.619547i
\(754\) 0 0
\(755\) −10.1185 4.19124i −0.368251 0.152535i
\(756\) 0 0
\(757\) 1.09998 + 2.65558i 0.0399793 + 0.0965185i 0.942608 0.333902i \(-0.108365\pi\)
−0.902629 + 0.430420i \(0.858365\pi\)
\(758\) 0 0
\(759\) 27.7025 + 27.7025i 1.00554 + 1.00554i
\(760\) 0 0
\(761\) 17.9432 17.9432i 0.650442 0.650442i −0.302658 0.953099i \(-0.597874\pi\)
0.953099 + 0.302658i \(0.0978739\pi\)
\(762\) 0 0
\(763\) −15.1740 + 6.28528i −0.549336 + 0.227542i
\(764\) 0 0
\(765\) 0.0279956 0.0675875i 0.00101218 0.00244363i
\(766\) 0 0
\(767\) −18.3056 −0.660976
\(768\) 0 0
\(769\) −2.98104 −0.107499 −0.0537495 0.998554i \(-0.517117\pi\)
−0.0537495 + 0.998554i \(0.517117\pi\)
\(770\) 0 0
\(771\) −1.36308 + 3.29077i −0.0490902 + 0.118514i
\(772\) 0 0
\(773\) 17.4049 7.20933i 0.626009 0.259302i −0.0470471 0.998893i \(-0.514981\pi\)
0.673056 + 0.739591i \(0.264981\pi\)
\(774\) 0 0
\(775\) 5.07886 5.07886i 0.182438 0.182438i
\(776\) 0 0
\(777\) 11.4688 + 11.4688i 0.411442 + 0.411442i
\(778\) 0 0
\(779\) −19.3531 46.7225i −0.693396 1.67401i
\(780\) 0 0
\(781\) 46.0056 + 19.0561i 1.64621 + 0.681882i
\(782\) 0 0
\(783\) 6.98779i 0.249723i
\(784\) 0 0
\(785\) 9.62716i 0.343608i
\(786\) 0 0
\(787\) −32.1235 13.3060i −1.14508 0.474307i −0.272198 0.962241i \(-0.587751\pi\)
−0.872881 + 0.487934i \(0.837751\pi\)
\(788\) 0 0
\(789\) −16.6727 40.2514i −0.593563 1.43299i
\(790\) 0 0
\(791\) −11.2130 11.2130i −0.398689 0.398689i
\(792\) 0 0
\(793\) −34.6525 + 34.6525i −1.23054 + 1.23054i
\(794\) 0 0
\(795\) −6.43957 + 2.66736i −0.228388 + 0.0946015i
\(796\) 0 0
\(797\) −6.84385 + 16.5225i −0.242422 + 0.585258i −0.997522 0.0703512i \(-0.977588\pi\)
0.755101 + 0.655609i \(0.227588\pi\)
\(798\) 0 0
\(799\) −51.2724 −1.81389
\(800\) 0 0
\(801\) −0.0406932 −0.00143782
\(802\) 0 0
\(803\) 3.42649 8.27229i 0.120918 0.291923i
\(804\) 0 0
\(805\) 7.50864 3.11018i 0.264645 0.109620i
\(806\) 0 0
\(807\) 18.0253 18.0253i 0.634521 0.634521i
\(808\) 0 0
\(809\) −10.5558 10.5558i −0.371123 0.371123i 0.496763 0.867886i \(-0.334522\pi\)
−0.867886 + 0.496763i \(0.834522\pi\)
\(810\) 0 0
\(811\) −1.78652 4.31304i −0.0627333 0.151451i 0.889404 0.457122i \(-0.151120\pi\)
−0.952138 + 0.305670i \(0.901120\pi\)
\(812\) 0 0
\(813\) −3.81894 1.58186i −0.133936 0.0554782i
\(814\) 0 0
\(815\) 2.83121i 0.0991728i
\(816\) 0 0
\(817\) 2.27601i 0.0796274i
\(818\) 0 0
\(819\) 0.339914 + 0.140797i 0.0118776 + 0.00491985i
\(820\) 0 0
\(821\) −10.5300 25.4218i −0.367501 0.887225i −0.994158 0.107931i \(-0.965578\pi\)
0.626658 0.779295i \(-0.284422\pi\)
\(822\) 0 0
\(823\) −28.1280 28.1280i −0.980479 0.980479i 0.0193343 0.999813i \(-0.493845\pi\)
−0.999813 + 0.0193343i \(0.993845\pi\)
\(824\) 0 0
\(825\) −21.3163 + 21.3163i −0.742138 + 0.742138i
\(826\) 0 0
\(827\) 14.6219 6.05658i 0.508453 0.210608i −0.113683 0.993517i \(-0.536265\pi\)
0.622136 + 0.782909i \(0.286265\pi\)
\(828\) 0 0
\(829\) −11.7436 + 28.3516i −0.407872 + 0.984691i 0.577824 + 0.816161i \(0.303902\pi\)
−0.985697 + 0.168530i \(0.946098\pi\)
\(830\) 0 0
\(831\) 17.2507 0.598419
\(832\) 0 0
\(833\) −6.46858 −0.224123
\(834\) 0 0
\(835\) −1.05902 + 2.55671i −0.0366490 + 0.0884786i
\(836\) 0 0
\(837\) −7.37881 + 3.05640i −0.255049 + 0.105645i
\(838\) 0 0
\(839\) −19.2678 + 19.2678i −0.665199 + 0.665199i −0.956601 0.291401i \(-0.905878\pi\)
0.291401 + 0.956601i \(0.405878\pi\)
\(840\) 0 0
\(841\) 19.2378 + 19.2378i 0.663372 + 0.663372i
\(842\) 0 0
\(843\) −18.5448 44.7710i −0.638715 1.54199i
\(844\) 0 0
\(845\) −12.5156 5.18414i −0.430551 0.178340i
\(846\) 0 0
\(847\) 6.90528i 0.237268i
\(848\) 0 0
\(849\) 8.89609i 0.305313i
\(850\) 0 0
\(851\) 22.0513 + 9.13394i 0.755908 + 0.313107i
\(852\) 0 0
\(853\) −5.53470 13.3619i −0.189504 0.457504i 0.800360 0.599520i \(-0.204642\pi\)
−0.989864 + 0.142016i \(0.954642\pi\)
\(854\) 0 0
\(855\) −0.0403999 0.0403999i −0.00138165 0.00138165i
\(856\) 0 0
\(857\) −16.2115 + 16.2115i −0.553775 + 0.553775i −0.927528 0.373753i \(-0.878071\pi\)
0.373753 + 0.927528i \(0.378071\pi\)
\(858\) 0 0
\(859\) −25.6351 + 10.6184i −0.874658 + 0.362295i −0.774423 0.632669i \(-0.781960\pi\)
−0.100235 + 0.994964i \(0.531960\pi\)
\(860\) 0 0
\(861\) 19.4936 47.0617i 0.664341 1.60386i
\(862\) 0 0
\(863\) 40.8195 1.38951 0.694755 0.719246i \(-0.255513\pi\)
0.694755 + 0.719246i \(0.255513\pi\)
\(864\) 0 0
\(865\) 9.12521 0.310267
\(866\) 0 0
\(867\) −7.09454 + 17.1277i −0.240943 + 0.581689i
\(868\) 0 0
\(869\) −44.1625 + 18.2927i −1.49811 + 0.620538i
\(870\) 0 0
\(871\) 24.9792 24.9792i 0.846389 0.846389i
\(872\) 0 0
\(873\) −0.0325505 0.0325505i −0.00110167 0.00110167i
\(874\) 0 0
\(875\) 4.94336 + 11.9343i 0.167116 + 0.403454i
\(876\) 0 0
\(877\) 43.8869 + 18.1786i 1.48196 + 0.613846i 0.969549 0.244896i \(-0.0787540\pi\)
0.512407 + 0.858743i \(0.328754\pi\)
\(878\) 0 0
\(879\) 37.8142i 1.27544i
\(880\) 0 0
\(881\) 0.849166i 0.0286091i 0.999898 + 0.0143046i \(0.00455344\pi\)
−0.999898 + 0.0143046i \(0.995447\pi\)
\(882\) 0 0
\(883\) −49.9223 20.6785i −1.68002 0.695886i −0.680692 0.732570i \(-0.738321\pi\)
−0.999327 + 0.0366838i \(0.988321\pi\)
\(884\) 0 0
\(885\) 1.09574 + 2.64534i 0.0368327 + 0.0889221i
\(886\) 0 0
\(887\) 17.8530 + 17.8530i 0.599446 + 0.599446i 0.940165 0.340719i \(-0.110671\pi\)
−0.340719 + 0.940165i \(0.610671\pi\)
\(888\) 0 0
\(889\) 7.42029 7.42029i 0.248868 0.248868i
\(890\) 0 0
\(891\) 30.7109 12.7209i 1.02885 0.426165i
\(892\) 0 0
\(893\) −15.3238 + 36.9950i −0.512792 + 1.23799i
\(894\) 0 0
\(895\) 12.5531 0.419604
\(896\) 0 0
\(897\) −64.3400 −2.14825
\(898\) 0 0
\(899\) 0.784527 1.89402i 0.0261655 0.0631690i
\(900\) 0 0
\(901\) −35.4521 + 14.6847i −1.18108 + 0.489220i
\(902\) 0 0
\(903\) 1.62107 1.62107i 0.0539457 0.0539457i
\(904\) 0 0
\(905\) 8.82413 + 8.82413i 0.293324 + 0.293324i
\(906\) 0 0
\(907\) −4.24679 10.2527i −0.141012 0.340434i 0.837558 0.546349i \(-0.183983\pi\)
−0.978570 + 0.205915i \(0.933983\pi\)
\(908\) 0 0
\(909\) −0.0441276 0.0182783i −0.00146362 0.000606252i
\(910\) 0 0
\(911\) 28.1022i 0.931067i −0.885030 0.465534i \(-0.845862\pi\)
0.885030 0.465534i \(-0.154138\pi\)
\(912\) 0 0
\(913\) 3.51032i 0.116174i
\(914\) 0 0
\(915\) 7.08185 + 2.93340i 0.234119 + 0.0969752i
\(916\) 0 0
\(917\) 6.26160 + 15.1168i 0.206776 + 0.499202i
\(918\) 0 0
\(919\) 5.17031 + 5.17031i 0.170553 + 0.170553i 0.787222 0.616669i \(-0.211518\pi\)
−0.616669 + 0.787222i \(0.711518\pi\)
\(920\) 0 0
\(921\) 26.3169 26.3169i 0.867172 0.867172i
\(922\) 0 0
\(923\) −75.5541 + 31.2955i −2.48689 + 1.03011i
\(924\) 0 0
\(925\) −7.02831 + 16.9678i −0.231089 + 0.557899i
\(926\) 0 0
\(927\) −0.190023 −0.00624116
\(928\) 0 0
\(929\) −21.6165 −0.709215 −0.354608 0.935015i \(-0.615386\pi\)
−0.354608 + 0.935015i \(0.615386\pi\)
\(930\) 0 0
\(931\) −1.93327 + 4.66733i −0.0633604 + 0.152965i
\(932\) 0 0
\(933\) 34.8848 14.4498i 1.14208 0.473064i
\(934\) 0 0
\(935\) 7.69664 7.69664i 0.251707 0.251707i
\(936\) 0 0
\(937\) 32.3267 + 32.3267i 1.05607 + 1.05607i 0.998332 + 0.0577344i \(0.0183876\pi\)
0.0577344 + 0.998332i \(0.481612\pi\)
\(938\) 0 0
\(939\) 13.1541 + 31.7569i 0.429269 + 1.03635i
\(940\) 0 0
\(941\) 23.9077 + 9.90288i 0.779368 + 0.322825i 0.736660 0.676263i \(-0.236402\pi\)
0.0427073 + 0.999088i \(0.486402\pi\)
\(942\) 0 0
\(943\) 74.9612i 2.44107i
\(944\) 0 0
\(945\) 6.95390i 0.226210i
\(946\) 0 0
\(947\) −21.6767 8.97880i −0.704399 0.291772i 0.00158544 0.999999i \(-0.499495\pi\)
−0.705985 + 0.708227i \(0.749495\pi\)
\(948\) 0 0
\(949\) 5.62726 + 13.5854i 0.182669 + 0.441001i
\(950\) 0 0
\(951\) −9.74220 9.74220i −0.315912 0.315912i
\(952\) 0 0
\(953\) 3.00440 3.00440i 0.0973222 0.0973222i −0.656769 0.754092i \(-0.728077\pi\)
0.754092 + 0.656769i \(0.228077\pi\)
\(954\) 0 0
\(955\) −5.34895 + 2.21561i −0.173088 + 0.0716954i
\(956\) 0 0
\(957\) −3.29271 + 7.94930i −0.106438 + 0.256965i
\(958\) 0 0
\(959\) −18.6010 −0.600657
\(960\) 0 0
\(961\) −28.6569 −0.924415
\(962\) 0 0
\(963\) 0.180210 0.435066i 0.00580720 0.0140198i
\(964\) 0 0
\(965\) 1.95682 0.810542i 0.0629923 0.0260923i
\(966\) 0 0
\(967\) −29.1087 + 29.1087i −0.936074 + 0.936074i −0.998076 0.0620023i \(-0.980251\pi\)
0.0620023 + 0.998076i \(0.480251\pi\)
\(968\) 0 0
\(969\) 26.4307 + 26.4307i 0.849075 + 0.849075i
\(970\) 0 0
\(971\) −0.258452 0.623957i −0.00829411 0.0200237i 0.919678 0.392673i \(-0.128450\pi\)
−0.927972 + 0.372649i \(0.878450\pi\)
\(972\) 0 0
\(973\) −25.7690 10.6739i −0.826117 0.342189i
\(974\) 0 0
\(975\) 49.5079i 1.58552i
\(976\) 0 0
\(977\) 33.1041i 1.05909i 0.848281 + 0.529546i \(0.177638\pi\)
−0.848281 + 0.529546i \(0.822362\pi\)
\(978\) 0 0
\(979\) −5.59374 2.31700i −0.178777 0.0740518i
\(980\) 0 0
\(981\) 0.0654936 + 0.158116i 0.00209105 + 0.00504824i
\(982\) 0 0
\(983\) −1.49880 1.49880i −0.0478044 0.0478044i 0.682800 0.730605i \(-0.260762\pi\)
−0.730605 + 0.682800i \(0.760762\pi\)
\(984\) 0 0
\(985\) 6.23264 6.23264i 0.198588 0.198588i
\(986\) 0 0
\(987\) −37.2636 + 15.4351i −1.18611 + 0.491304i
\(988\) 0 0
\(989\) 1.29104 3.11684i 0.0410526 0.0991098i
\(990\) 0 0
\(991\) 25.9387 0.823970 0.411985 0.911191i \(-0.364836\pi\)
0.411985 + 0.911191i \(0.364836\pi\)
\(992\) 0 0
\(993\) 12.0895 0.383649
\(994\) 0 0
\(995\) −4.81094 + 11.6146i −0.152517 + 0.368209i
\(996\) 0 0
\(997\) −5.97395 + 2.47449i −0.189197 + 0.0783679i −0.475270 0.879840i \(-0.657650\pi\)
0.286074 + 0.958208i \(0.407650\pi\)
\(998\) 0 0
\(999\) 14.4406 14.4406i 0.456881 0.456881i
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 1024.2.g.e.129.3 yes 16
4.3 odd 2 inner 1024.2.g.e.129.2 yes 16
8.3 odd 2 1024.2.g.b.129.3 yes 16
8.5 even 2 1024.2.g.b.129.2 16
16.3 odd 4 1024.2.g.c.641.2 yes 16
16.5 even 4 1024.2.g.h.641.2 yes 16
16.11 odd 4 1024.2.g.h.641.3 yes 16
16.13 even 4 1024.2.g.c.641.3 yes 16
32.3 odd 8 1024.2.g.c.385.2 yes 16
32.5 even 8 inner 1024.2.g.e.897.3 yes 16
32.11 odd 8 1024.2.g.b.897.3 yes 16
32.13 even 8 1024.2.g.h.385.2 yes 16
32.19 odd 8 1024.2.g.h.385.3 yes 16
32.21 even 8 1024.2.g.b.897.2 yes 16
32.27 odd 8 inner 1024.2.g.e.897.2 yes 16
32.29 even 8 1024.2.g.c.385.3 yes 16
64.5 even 16 4096.2.a.n.1.5 8
64.27 odd 16 4096.2.a.n.1.6 8
64.37 even 16 4096.2.a.o.1.4 8
64.59 odd 16 4096.2.a.o.1.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1024.2.g.b.129.2 16 8.5 even 2
1024.2.g.b.129.3 yes 16 8.3 odd 2
1024.2.g.b.897.2 yes 16 32.21 even 8
1024.2.g.b.897.3 yes 16 32.11 odd 8
1024.2.g.c.385.2 yes 16 32.3 odd 8
1024.2.g.c.385.3 yes 16 32.29 even 8
1024.2.g.c.641.2 yes 16 16.3 odd 4
1024.2.g.c.641.3 yes 16 16.13 even 4
1024.2.g.e.129.2 yes 16 4.3 odd 2 inner
1024.2.g.e.129.3 yes 16 1.1 even 1 trivial
1024.2.g.e.897.2 yes 16 32.27 odd 8 inner
1024.2.g.e.897.3 yes 16 32.5 even 8 inner
1024.2.g.h.385.2 yes 16 32.13 even 8
1024.2.g.h.385.3 yes 16 32.19 odd 8
1024.2.g.h.641.2 yes 16 16.5 even 4
1024.2.g.h.641.3 yes 16 16.11 odd 4
4096.2.a.n.1.5 8 64.5 even 16
4096.2.a.n.1.6 8 64.27 odd 16
4096.2.a.o.1.3 8 64.59 odd 16
4096.2.a.o.1.4 8 64.37 even 16