Properties

Label 1024.2.g.c.641.3
Level $1024$
Weight $2$
Character 1024.641
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 641.3
Root \(1.50947 - 0.849413i\) of defining polynomial
Character \(\chi\) \(=\) 1024.641
Dual form 1024.2.g.c.385.3

$q$-expansion

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

Character values

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

\(n\) \(5\) \(1023\)
\(\chi(n)\) \(e\left(\frac{3}{8}\right)\) \(1\)

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
\(3\) 1.59352 + 0.660056i 0.920017 + 0.381083i 0.791883 0.610673i \(-0.209101\pi\)
0.128134 + 0.991757i \(0.459101\pi\)
\(4\) 0 0
\(5\) 0.212292 + 0.512517i 0.0949397 + 0.229205i 0.964214 0.265125i \(-0.0854134\pi\)
−0.869274 + 0.494330i \(0.835413\pi\)
\(6\) 0 0
\(7\) 1.69883 1.69883i 0.642096 0.642096i −0.308974 0.951070i \(-0.599986\pi\)
0.951070 + 0.308974i \(0.0999857\pi\)
\(8\) 0 0
\(9\) −0.0177021 0.0177021i −0.00590068 0.00590068i
\(10\) 0 0
\(11\) 3.44127 1.42542i 1.03758 0.429781i 0.202139 0.979357i \(-0.435211\pi\)
0.835444 + 0.549576i \(0.185211\pi\)
\(12\) 0 0
\(13\) 2.34094 5.65154i 0.649261 1.56746i −0.164577 0.986364i \(-0.552626\pi\)
0.813839 0.581091i \(-0.197374\pi\)
\(14\) 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
\(19\) −1.57436 + 3.80083i −0.361182 + 0.871970i 0.633946 + 0.773377i \(0.281434\pi\)
−0.995128 + 0.0985928i \(0.968566\pi\)
\(20\) 0 0
\(21\) 3.82843 1.58579i 0.835431 0.346047i
\(22\) 0 0
\(23\) −4.31195 4.31195i −0.899104 0.899104i 0.0962527 0.995357i \(-0.469314\pi\)
−0.995357 + 0.0962527i \(0.969314\pi\)
\(24\) 0 0
\(25\) 3.31793 3.31793i 0.663586 0.663586i
\(26\) 0 0
\(27\) −1.99669 4.82044i −0.384263 0.927694i
\(28\) 0 0
\(29\) −1.23733 0.512517i −0.229766 0.0951721i 0.264830 0.964295i \(-0.414684\pi\)
−0.494596 + 0.869123i \(0.664684\pi\)
\(30\) 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
\(35\) 1.23132 + 0.510031i 0.208132 + 0.0862110i
\(36\) 0 0
\(37\) −1.49785 3.61614i −0.246245 0.594489i 0.751634 0.659581i \(-0.229266\pi\)
−0.997879 + 0.0650915i \(0.979266\pi\)
\(38\) 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 \(0.159551\pi\)
0.480518 + 0.876985i \(0.340449\pi\)
\(42\) 0 0
\(43\) −0.511123 + 0.211714i −0.0779456 + 0.0322861i −0.421316 0.906914i \(-0.638432\pi\)
0.343370 + 0.939200i \(0.388432\pi\)
\(44\) 0 0
\(45\) 0.00531461 0.0128306i 0.000792255 0.00191267i
\(46\) 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
\(51\) −3.47696 + 8.39412i −0.486872 + 1.17541i
\(52\) 0 0
\(53\) 6.73012 2.78771i 0.924454 0.382921i 0.130882 0.991398i \(-0.458219\pi\)
0.793572 + 0.608477i \(0.208219\pi\)
\(54\) 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
\(59\) −1.14517 2.76469i −0.149089 0.359933i 0.831637 0.555319i \(-0.187404\pi\)
−0.980726 + 0.195387i \(0.937404\pi\)
\(60\) 0 0
\(61\) 7.40138 + 3.06575i 0.947650 + 0.392529i 0.802347 0.596858i \(-0.203584\pi\)
0.145303 + 0.989387i \(0.453584\pi\)
\(62\) 0 0
\(63\) −0.0601454 −0.00757761
\(64\) 0 0
\(65\) 3.39347 0.420909
\(66\) 0 0
\(67\) −5.33529 2.20995i −0.651810 0.269988i 0.0321778 0.999482i \(-0.489756\pi\)
−0.683987 + 0.729494i \(0.739756\pi\)
\(68\) 0 0
\(69\) −4.02503 9.71729i −0.484557 1.16982i
\(70\) 0 0
\(71\) −9.45315 + 9.45315i −1.12188 + 1.12188i −0.130424 + 0.991458i \(0.541634\pi\)
−0.991458 + 0.130424i \(0.958366\pi\)
\(72\) 0 0
\(73\) 1.69977 + 1.69977i 0.198944 + 0.198944i 0.799547 0.600603i \(-0.205073\pi\)
−0.600603 + 0.799547i \(0.705073\pi\)
\(74\) 0 0
\(75\) 7.47719 3.09715i 0.863391 0.357628i
\(76\) 0 0
\(77\) 3.42458 8.26768i 0.390267 0.942189i
\(78\) 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
\(83\) 0.360647 0.870678i 0.0395861 0.0955693i −0.902850 0.429956i \(-0.858529\pi\)
0.942436 + 0.334387i \(0.108529\pi\)
\(84\) 0 0
\(85\) −2.69977 + 1.11828i −0.292832 + 0.121295i
\(86\) 0 0
\(87\) −1.63341 1.63341i −0.175120 0.175120i
\(88\) 0 0
\(89\) 1.14939 1.14939i 0.121835 0.121835i −0.643560 0.765396i \(-0.722543\pi\)
0.765396 + 0.643560i \(0.222543\pi\)
\(90\) 0 0
\(91\) −5.62413 13.5778i −0.589569 1.42334i
\(92\) 0 0
\(93\) −2.43925 1.01037i −0.252938 0.104770i
\(94\) 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
\(99\) −0.0861505 0.0356847i −0.00865845 0.00358645i
\(100\) 0 0
\(101\) 0.730123 + 1.76267i 0.0726500 + 0.175393i 0.956033 0.293259i \(-0.0947399\pi\)
−0.883383 + 0.468652i \(0.844740\pi\)
\(102\) 0 0
\(103\) 5.36725 5.36725i 0.528851 0.528851i −0.391379 0.920230i \(-0.628002\pi\)
0.920230 + 0.391379i \(0.128002\pi\)
\(104\) 0 0
\(105\) 1.62549 + 1.62549i 0.158631 + 0.158631i
\(106\) 0 0
\(107\) −17.3787 + 7.19848i −1.68006 + 0.695904i −0.999329 0.0366299i \(-0.988338\pi\)
−0.680731 + 0.732534i \(0.738338\pi\)
\(108\) 0 0
\(109\) −2.61614 + 6.31591i −0.250580 + 0.604954i −0.998251 0.0591151i \(-0.981172\pi\)
0.747671 + 0.664070i \(0.231172\pi\)
\(110\) 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
\(115\) 1.29456 3.12534i 0.120718 0.291440i
\(116\) 0 0
\(117\) −0.141483 + 0.0586043i −0.0130801 + 0.00541797i
\(118\) 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
\(123\) 8.11387 + 19.5886i 0.731603 + 1.76625i
\(124\) 0 0
\(125\) 4.96745 + 2.05758i 0.444302 + 0.184036i
\(126\) 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
\(131\) 6.29212 + 2.60628i 0.549745 + 0.227712i 0.640226 0.768186i \(-0.278841\pi\)
−0.0904813 + 0.995898i \(0.528841\pi\)
\(132\) 0 0
\(133\) 3.78239 + 9.13151i 0.327975 + 0.791802i
\(134\) 0 0
\(135\) 2.04668 2.04668i 0.176150 0.176150i
\(136\) 0 0
\(137\) −5.47465 5.47465i −0.467731 0.467731i 0.433448 0.901179i \(-0.357297\pi\)
−0.901179 + 0.433448i \(0.857297\pi\)
\(138\) 0 0
\(139\) 10.7259 4.44282i 0.909760 0.376835i 0.121795 0.992555i \(-0.461135\pi\)
0.787965 + 0.615721i \(0.211135\pi\)
\(140\) 0 0
\(141\) −6.42458 + 15.5103i −0.541048 + 1.30620i
\(142\) 0 0
\(143\) 22.7853i 1.90541i
\(144\) 0 0
\(145\) 0.742954i 0.0616990i
\(146\) 0 0
\(147\) −0.810533 + 1.95680i −0.0668516 + 0.161394i
\(148\) 0 0
\(149\) 4.76267 1.97276i 0.390173 0.161615i −0.178968 0.983855i \(-0.557276\pi\)
0.569141 + 0.822240i \(0.307276\pi\)
\(150\) 0 0
\(151\) −13.9603 13.9603i −1.13607 1.13607i −0.989148 0.146924i \(-0.953063\pi\)
−0.146924 0.989148i \(-0.546937\pi\)
\(152\) 0 0
\(153\) 0.0932487 0.0932487i 0.00753871 0.00753871i
\(154\) 0 0
\(155\) −0.324962 0.784527i −0.0261016 0.0630148i
\(156\) 0 0
\(157\) −16.0332 6.64117i −1.27959 0.530023i −0.363723 0.931507i \(-0.618494\pi\)
−0.915866 + 0.401484i \(0.868494\pi\)
\(158\) 0 0
\(159\) 12.5646 0.996438
\(160\) 0 0
\(161\) −14.6505 −1.15462
\(162\) 0 0
\(163\) −4.71513 1.95307i −0.369317 0.152976i 0.190303 0.981725i \(-0.439053\pi\)
−0.559620 + 0.828749i \(0.689053\pi\)
\(164\) 0 0
\(165\) 1.36388 + 3.29271i 0.106178 + 0.256337i
\(166\) 0 0
\(167\) −3.52743 + 3.52743i −0.272960 + 0.272960i −0.830291 0.557330i \(-0.811826\pi\)
0.557330 + 0.830291i \(0.311826\pi\)
\(168\) 0 0
\(169\) −17.2675 17.2675i −1.32827 1.32827i
\(170\) 0 0
\(171\) 0.0951518 0.0394132i 0.00727644 0.00301400i
\(172\) 0 0
\(173\) 6.29491 15.1973i 0.478593 1.15543i −0.481676 0.876350i \(-0.659972\pi\)
0.960269 0.279077i \(-0.0900284\pi\)
\(174\) 0 0
\(175\) 11.2732i 0.852171i
\(176\) 0 0
\(177\) 5.16146i 0.387959i
\(178\) 0 0
\(179\) 8.65959 20.9061i 0.647248 1.56260i −0.169455 0.985538i \(-0.554201\pi\)
0.816704 0.577057i \(-0.195799\pi\)
\(180\) 0 0
\(181\) −20.7830 + 8.60862i −1.54479 + 0.639874i −0.982365 0.186974i \(-0.940132\pi\)
−0.562427 + 0.826847i \(0.690132\pi\)
\(182\) 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
\(187\) 7.50866 + 18.1275i 0.549088 + 1.32562i
\(188\) 0 0
\(189\) −11.5811 4.79706i −0.842403 0.348935i
\(190\) 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
\(195\) 5.40755 + 2.23988i 0.387243 + 0.160401i
\(196\) 0 0
\(197\) 6.08042 + 14.6794i 0.433212 + 1.04587i 0.978245 + 0.207452i \(0.0665170\pi\)
−0.545033 + 0.838414i \(0.683483\pi\)
\(198\) 0 0
\(199\) −16.0244 + 16.0244i −1.13594 + 1.13594i −0.146771 + 0.989171i \(0.546888\pi\)
−0.989171 + 0.146771i \(0.953112\pi\)
\(200\) 0 0
\(201\) −7.04318 7.04318i −0.496788 0.496788i
\(202\) 0 0
\(203\) −2.97268 + 1.23132i −0.208641 + 0.0864220i
\(204\) 0 0
\(205\) −2.60964 + 6.30023i −0.182265 + 0.440027i
\(206\) 0 0
\(207\) 0.152661i 0.0106107i
\(208\) 0 0
\(209\) 15.3238i 1.05997i
\(210\) 0 0
\(211\) 5.50184 13.2826i 0.378762 0.914413i −0.613436 0.789744i \(-0.710213\pi\)
0.992198 0.124669i \(-0.0397868\pi\)
\(212\) 0 0
\(213\) −21.3033 + 8.82413i −1.45968 + 0.604620i
\(214\) 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
\(219\) 1.58667 + 3.83056i 0.107217 + 0.258845i
\(220\) 0 0
\(221\) 29.7705 + 12.3313i 2.00258 + 0.829495i
\(222\) 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
\(227\) 5.19652 + 2.15247i 0.344905 + 0.142864i 0.548410 0.836210i \(-0.315233\pi\)
−0.203505 + 0.979074i \(0.565233\pi\)
\(228\) 0 0
\(229\) 5.04072 + 12.1694i 0.333100 + 0.804175i 0.998343 + 0.0575475i \(0.0183281\pi\)
−0.665243 + 0.746627i \(0.731672\pi\)
\(230\) 0 0
\(231\) 10.9143 10.9143i 0.718105 0.718105i
\(232\) 0 0
\(233\) −1.74984 1.74984i −0.114636 0.114636i 0.647462 0.762098i \(-0.275831\pi\)
−0.762098 + 0.647462i \(0.775831\pi\)
\(234\) 0 0
\(235\) −4.98853 + 2.06632i −0.325416 + 0.134792i
\(236\) 0 0
\(237\) −8.47062 + 20.4499i −0.550226 + 1.32836i
\(238\) 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
\(243\) −0.0995586 + 0.240356i −0.00638669 + 0.0154188i
\(244\) 0 0
\(245\) −0.629359 + 0.260689i −0.0402083 + 0.0166548i
\(246\) 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
\(251\) 3.77199 + 9.10639i 0.238086 + 0.574790i 0.997085 0.0763018i \(-0.0243113\pi\)
−0.758999 + 0.651092i \(0.774311\pi\)
\(252\) 0 0
\(253\) −20.9850 8.69226i −1.31931 0.546478i
\(254\) 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
\(259\) −8.68778 3.59860i −0.539832 0.223606i
\(260\) 0 0
\(261\) 0.0128306 + 0.0309758i 0.000794194 + 0.00191736i
\(262\) 0 0
\(263\) −17.8612 + 17.8612i −1.10137 + 1.10137i −0.107120 + 0.994246i \(0.534163\pi\)
−0.994246 + 0.107120i \(0.965837\pi\)
\(264\) 0 0
\(265\) 2.85750 + 2.85750i 0.175535 + 0.175535i
\(266\) 0 0
\(267\) 2.59024 1.07291i 0.158520 0.0656611i
\(268\) 0 0
\(269\) −5.65583 + 13.6544i −0.344842 + 0.832523i 0.652370 + 0.757901i \(0.273775\pi\)
−0.997212 + 0.0746220i \(0.976225\pi\)
\(270\) 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
\(275\) 6.68845 16.1473i 0.403329 0.973722i
\(276\) 0 0
\(277\) 9.24018 3.82741i 0.555189 0.229967i −0.0874063 0.996173i \(-0.527858\pi\)
0.642595 + 0.766206i \(0.277858\pi\)
\(278\) 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 \(0.566290\pi\)
−0.978393 + 0.206754i \(0.933710\pi\)
\(282\) 0 0
\(283\) −1.97378 4.76511i −0.117329 0.283257i 0.854295 0.519788i \(-0.173989\pi\)
−0.971624 + 0.236532i \(0.923989\pi\)
\(284\) 0 0
\(285\) −3.63674 1.50639i −0.215422 0.0892307i
\(286\) 0 0
\(287\) 29.5333 1.74329
\(288\) 0 0
\(289\) −10.7484 −0.632259
\(290\) 0 0
\(291\) −2.93015 1.21371i −0.171769 0.0711488i
\(292\) 0 0
\(293\) −8.38983 20.2548i −0.490139 1.18330i −0.954649 0.297733i \(-0.903770\pi\)
0.464510 0.885568i \(-0.346230\pi\)
\(294\) 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
\(299\) −34.4632 + 14.2751i −1.99306 + 0.825552i
\(300\) 0 0
\(301\) −0.508644 + 1.22798i −0.0293178 + 0.0707793i
\(302\) 0 0
\(303\) 3.29077i 0.189050i
\(304\) 0 0
\(305\) 4.44417i 0.254472i
\(306\) 0 0
\(307\) −8.25751 + 19.9354i −0.471281 + 1.13777i 0.492317 + 0.870416i \(0.336150\pi\)
−0.963598 + 0.267356i \(0.913850\pi\)
\(308\) 0 0
\(309\) 12.0955 5.01011i 0.688088 0.285015i
\(310\) 0 0
\(311\) −15.4798 15.4798i −0.877779 0.877779i 0.115526 0.993304i \(-0.463145\pi\)
−0.993304 + 0.115526i \(0.963145\pi\)
\(312\) 0 0
\(313\) 14.0918 14.0918i 0.796516 0.796516i −0.186028 0.982544i \(-0.559562\pi\)
0.982544 + 0.186028i \(0.0595616\pi\)
\(314\) 0 0
\(315\) −0.0127684 0.0308256i −0.000719416 0.00173682i
\(316\) 0 0
\(317\) 7.37983 + 3.05683i 0.414492 + 0.171688i 0.580177 0.814490i \(-0.302983\pi\)
−0.165685 + 0.986179i \(0.552983\pi\)
\(318\) 0 0
\(319\) −4.98853 −0.279304
\(320\) 0 0
\(321\) −32.4446 −1.81088
\(322\) 0 0
\(323\) −20.0215 8.29319i −1.11403 0.461445i
\(324\) 0 0
\(325\) −10.9843 26.5185i −0.609300 1.47098i
\(326\) 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
\(331\) 6.47564 2.68230i 0.355934 0.147433i −0.197550 0.980293i \(-0.563298\pi\)
0.553483 + 0.832860i \(0.313298\pi\)
\(332\) 0 0
\(333\) −0.0374980 + 0.0905281i −0.00205488 + 0.00496091i
\(334\) 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
\(339\) −4.35667 + 10.5179i −0.236622 + 0.571255i
\(340\) 0 0
\(341\) −5.26768 + 2.18194i −0.285261 + 0.118159i
\(342\) 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
\(347\) −4.39235 10.6041i −0.235794 0.569256i 0.761046 0.648698i \(-0.224686\pi\)
−0.996840 + 0.0794417i \(0.974686\pi\)
\(348\) 0 0
\(349\) 19.7580 + 8.18404i 1.05762 + 0.438081i 0.842607 0.538529i \(-0.181020\pi\)
0.215015 + 0.976611i \(0.431020\pi\)
\(350\) 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
\(355\) −6.85172 2.83808i −0.363652 0.150629i
\(356\) 0 0
\(357\) 8.35341 + 20.1669i 0.442109 + 1.06735i
\(358\) 0 0
\(359\) −6.77460 + 6.77460i −0.357550 + 0.357550i −0.862909 0.505359i \(-0.831360\pi\)
0.505359 + 0.862909i \(0.331360\pi\)
\(360\) 0 0
\(361\) 1.46732 + 1.46732i 0.0772274 + 0.0772274i
\(362\) 0 0
\(363\) 4.58008 1.89713i 0.240392 0.0995737i
\(364\) 0 0
\(365\) −0.510316 + 1.23201i −0.0267112 + 0.0644864i
\(366\) 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
\(371\) 6.69748 16.1691i 0.347716 0.839460i
\(372\) 0 0
\(373\) −6.81560 + 2.82311i −0.352898 + 0.146175i −0.552088 0.833786i \(-0.686169\pi\)
0.199190 + 0.979961i \(0.436169\pi\)
\(374\) 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
\(379\) −5.30491 12.8072i −0.272495 0.657861i 0.727094 0.686538i \(-0.240870\pi\)
−0.999589 + 0.0286774i \(0.990870\pi\)
\(380\) 0 0
\(381\) −6.96030 2.88305i −0.356587 0.147703i
\(382\) 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
\(387\) 0.0127957 + 0.00530016i 0.000650442 + 0.000269422i
\(388\) 0 0
\(389\) 7.93103 + 19.1472i 0.402119 + 0.970801i 0.987151 + 0.159791i \(0.0510820\pi\)
−0.585032 + 0.811010i \(0.698918\pi\)
\(390\) 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
\(395\) −6.57723 + 2.72438i −0.330936 + 0.137078i
\(396\) 0 0
\(397\) −2.65594 + 6.41201i −0.133298 + 0.321810i −0.976409 0.215930i \(-0.930722\pi\)
0.843111 + 0.537740i \(0.180722\pi\)
\(398\) 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
\(403\) −3.58336 + 8.65100i −0.178500 + 0.430937i
\(404\) 0 0
\(405\) 4.57384 1.89455i 0.227276 0.0941408i
\(406\) 0 0
\(407\) −10.3090 10.3090i −0.511000 0.511000i
\(408\) 0 0
\(409\) 0.106839 0.106839i 0.00528285 0.00528285i −0.704460 0.709743i \(-0.748811\pi\)
0.709743 + 0.704460i \(0.248811\pi\)
\(410\) 0 0
\(411\) −5.11037 12.3375i −0.252076 0.608565i
\(412\) 0 0
\(413\) −6.64219 2.75128i −0.326841 0.135382i
\(414\) 0 0
\(415\) 0.522800 0.0256632
\(416\) 0 0
\(417\) 20.0244 0.980599
\(418\) 0 0
\(419\) 18.7318 + 7.75898i 0.915110 + 0.379051i 0.790011 0.613093i \(-0.210075\pi\)
0.125099 + 0.992144i \(0.460075\pi\)
\(420\) 0 0
\(421\) −5.08471 12.2756i −0.247814 0.598275i 0.750204 0.661206i \(-0.229955\pi\)
−0.998018 + 0.0629310i \(0.979955\pi\)
\(422\) 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
\(427\) 17.7819 7.36548i 0.860524 0.356441i
\(428\) 0 0
\(429\) 15.0396 36.3088i 0.726118 1.75300i
\(430\) 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
\(435\) 0.490391 1.18391i 0.0235125 0.0567641i
\(436\) 0 0
\(437\) 23.1775 9.60045i 1.10873 0.459252i
\(438\) 0 0
\(439\) −5.35602 5.35602i −0.255629 0.255629i 0.567645 0.823274i \(-0.307855\pi\)
−0.823274 + 0.567645i \(0.807855\pi\)
\(440\) 0 0
\(441\) 0.0217377 0.0217377i 0.00103513 0.00103513i
\(442\) 0 0
\(443\) 7.13988 + 17.2372i 0.339226 + 0.818964i 0.997790 + 0.0664397i \(0.0211640\pi\)
−0.658564 + 0.752524i \(0.728836\pi\)
\(444\) 0 0
\(445\) 0.833089 + 0.345077i 0.0394922 + 0.0163582i
\(446\) 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
\(451\) 42.3026 + 17.5223i 1.99195 + 0.825093i
\(452\) 0 0
\(453\) −13.0314 31.4605i −0.612267 1.47814i
\(454\) 0 0
\(455\) 5.76492 5.76492i 0.270264 0.270264i
\(456\) 0 0
\(457\) −9.65659 9.65659i −0.451716 0.451716i 0.444207 0.895924i \(-0.353485\pi\)
−0.895924 + 0.444207i \(0.853485\pi\)
\(458\) 0 0
\(459\) 25.3925 10.5179i 1.18522 0.490935i
\(460\) 0 0
\(461\) 9.64976 23.2966i 0.449434 1.08503i −0.523100 0.852271i \(-0.675225\pi\)
0.972534 0.232759i \(-0.0747753\pi\)
\(462\) 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
\(467\) −9.79289 + 23.6421i −0.453161 + 1.09403i 0.517953 + 0.855409i \(0.326694\pi\)
−0.971114 + 0.238618i \(0.923306\pi\)
\(468\) 0 0
\(469\) −12.8181 + 5.30941i −0.591883 + 0.245166i
\(470\) 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
\(475\) 7.38728 + 17.8345i 0.338952 + 0.818302i
\(476\) 0 0
\(477\) −0.168485 0.0697888i −0.00771441 0.00319541i
\(478\) 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
\(483\) −23.3458 9.67016i −1.06227 0.440008i
\(484\) 0 0
\(485\) −0.390361 0.942415i −0.0177254 0.0427929i
\(486\) 0 0
\(487\) 15.4256 15.4256i 0.699001 0.699001i −0.265194 0.964195i \(-0.585436\pi\)
0.964195 + 0.265194i \(0.0854361\pi\)
\(488\) 0 0
\(489\) −6.22450 6.22450i −0.281481 0.281481i
\(490\) 0 0
\(491\) 28.8199 11.9376i 1.30062 0.538736i 0.378489 0.925606i \(-0.376444\pi\)
0.922134 + 0.386870i \(0.126444\pi\)
\(492\) 0 0
\(493\) 2.69977 6.51783i 0.121592 0.293548i
\(494\) 0 0
\(495\) 0.0517292i 0.00232505i
\(496\) 0 0
\(497\) 32.1185i 1.44071i
\(498\) 0 0
\(499\) −14.6102 + 35.2722i −0.654043 + 1.57900i 0.152814 + 0.988255i \(0.451166\pi\)
−0.806858 + 0.590746i \(0.798834\pi\)
\(500\) 0 0
\(501\) −7.94930 + 3.29271i −0.355149 + 0.147107i
\(502\) 0 0
\(503\) 2.18240 + 2.18240i 0.0973084 + 0.0973084i 0.754085 0.656777i \(-0.228081\pi\)
−0.656777 + 0.754085i \(0.728081\pi\)
\(504\) 0 0
\(505\) −0.748402 + 0.748402i −0.0333034 + 0.0333034i
\(506\) 0 0
\(507\) −16.1185 38.9135i −0.715848 1.72821i
\(508\) 0 0
\(509\) 7.44742 + 3.08482i 0.330101 + 0.136732i 0.541578 0.840651i \(-0.317827\pi\)
−0.211477 + 0.977383i \(0.567827\pi\)
\(510\) 0 0
\(511\) 5.77524 0.255482
\(512\) 0 0
\(513\) 21.4652 0.947710
\(514\) 0 0
\(515\) 3.89023 + 1.61139i 0.171424 + 0.0710061i
\(516\) 0 0
\(517\) 13.8742 + 33.4953i 0.610187 + 1.47312i
\(518\) 0 0
\(519\) 20.0621 20.0621i 0.880627 0.880627i
\(520\) 0 0
\(521\) 6.94553 + 6.94553i 0.304289 + 0.304289i 0.842689 0.538400i \(-0.180971\pi\)
−0.538400 + 0.842689i \(0.680971\pi\)
\(522\) 0 0
\(523\) 2.24673 0.930625i 0.0982425 0.0406934i −0.333021 0.942919i \(-0.608068\pi\)
0.431263 + 0.902226i \(0.358068\pi\)
\(524\) 0 0
\(525\) 7.44092 17.9640i 0.324748 0.784012i
\(526\) 0 0
\(527\) 8.06341i 0.351248i
\(528\) 0 0
\(529\) 14.1859i 0.616777i
\(530\) 0 0
\(531\) −0.0286688 + 0.0692127i −0.00124412 + 0.00300358i
\(532\) 0 0
\(533\) 69.4727 28.7765i 3.00920 1.24645i
\(534\) 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
\(539\) 1.75038 + 4.22580i 0.0753944 + 0.182018i
\(540\) 0 0
\(541\) −25.4120 10.5260i −1.09255 0.452548i −0.237653 0.971350i \(-0.576378\pi\)
−0.854894 + 0.518802i \(0.826378\pi\)
\(542\) 0 0
\(543\) −38.8003 −1.66508
\(544\) 0 0
\(545\) −3.79240 −0.162448
\(546\) 0 0
\(547\) −2.52591 1.04627i −0.108000 0.0447351i 0.328029 0.944668i \(-0.393616\pi\)
−0.436029 + 0.899932i \(0.643616\pi\)
\(548\) 0 0
\(549\) −0.0767495 0.185290i −0.00327559 0.00790798i
\(550\) 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
\(555\) 3.46002 1.43319i 0.146870 0.0608354i
\(556\) 0 0
\(557\) 8.61495 20.7983i 0.365027 0.881254i −0.629522 0.776983i \(-0.716749\pi\)
0.994549 0.104271i \(-0.0332509\pi\)
\(558\) 0 0
\(559\) 3.38425i 0.143138i
\(560\) 0 0
\(561\) 33.8426i 1.42884i
\(562\) 0 0
\(563\) 8.66534 20.9200i 0.365201 0.881672i −0.629321 0.777145i \(-0.716667\pi\)
0.994522 0.104527i \(-0.0333329\pi\)
\(564\) 0 0
\(565\) −3.38285 + 1.40122i −0.142317 + 0.0589498i
\(566\) 0 0
\(567\) −15.1608 15.1608i −0.636693 0.636693i
\(568\) 0 0
\(569\) 0.276633 0.276633i 0.0115970 0.0115970i −0.701284 0.712882i \(-0.747390\pi\)
0.712882 + 0.701284i \(0.247390\pi\)
\(570\) 0 0
\(571\) −13.2460 31.9788i −0.554329 1.33827i −0.914198 0.405267i \(-0.867179\pi\)
0.359869 0.933003i \(-0.382821\pi\)
\(572\) 0 0
\(573\) 16.6309 + 6.88876i 0.694767 + 0.287782i
\(574\) 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
\(579\) −6.08413 2.52013i −0.252848 0.104733i
\(580\) 0 0
\(581\) −0.866455 2.09181i −0.0359466 0.0867828i
\(582\) 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
\(587\) −32.7941 + 13.5838i −1.35356 + 0.560662i −0.937281 0.348575i \(-0.886665\pi\)
−0.416277 + 0.909238i \(0.636665\pi\)
\(588\) 0 0
\(589\) 2.40992 5.81806i 0.0992990 0.239729i
\(590\) 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
\(595\) −2.68668 + 6.48622i −0.110143 + 0.265909i
\(596\) 0 0
\(597\) −36.1122 + 14.9582i −1.47797 + 0.612197i
\(598\) 0 0
\(599\) 11.9077 + 11.9077i 0.486534 + 0.486534i 0.907211 0.420677i \(-0.138207\pi\)
−0.420677 + 0.907211i \(0.638207\pi\)
\(600\) 0 0
\(601\) 1.86501 1.86501i 0.0760755 0.0760755i −0.668045 0.744121i \(-0.732869\pi\)
0.744121 + 0.668045i \(0.232869\pi\)
\(602\) 0 0
\(603\) 0.0553250 + 0.133566i 0.00225301 + 0.00543924i
\(604\) 0 0
\(605\) 1.47308 + 0.610169i 0.0598891 + 0.0248069i
\(606\) 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
\(611\) 55.0087 + 22.7853i 2.22541 + 0.921796i
\(612\) 0 0
\(613\) −9.67657 23.3613i −0.390833 0.943555i −0.989759 0.142750i \(-0.954405\pi\)
0.598926 0.800805i \(-0.295595\pi\)
\(614\) 0 0
\(615\) −8.31700 + 8.31700i −0.335374 + 0.335374i
\(616\) 0 0
\(617\) 8.91738 + 8.91738i 0.359000 + 0.359000i 0.863444 0.504444i \(-0.168303\pi\)
−0.504444 + 0.863444i \(0.668303\pi\)
\(618\) 0 0
\(619\) 5.56774 2.30623i 0.223786 0.0926953i −0.267973 0.963426i \(-0.586354\pi\)
0.491760 + 0.870731i \(0.336354\pi\)
\(620\) 0 0
\(621\) −12.1759 + 29.3951i −0.488601 + 1.17959i
\(622\) 0 0
\(623\) 3.90524i 0.156460i
\(624\) 0 0
\(625\) 20.4786i 0.819143i
\(626\) 0 0
\(627\) −10.1146 + 24.4187i −0.403937 + 0.975191i
\(628\) 0 0
\(629\) 19.0486 7.89020i 0.759519 0.314603i
\(630\) 0 0
\(631\) −33.4932 33.4932i −1.33334 1.33334i −0.902361 0.430981i \(-0.858167\pi\)
−0.430981 0.902361i \(-0.641833\pi\)
\(632\) 0 0
\(633\) 17.5345 17.5345i 0.696935 0.696935i
\(634\) 0 0
\(635\) −0.927266 2.23862i −0.0367974 0.0888369i
\(636\) 0 0
\(637\) 6.93995 + 2.87462i 0.274971 + 0.113897i
\(638\) 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
\(643\) −31.4666 13.0339i −1.24092 0.514006i −0.336918 0.941534i \(-0.609384\pi\)
−0.904002 + 0.427528i \(0.859384\pi\)
\(644\) 0 0
\(645\) −0.202574 0.489057i −0.00797635 0.0192566i
\(646\) 0 0
\(647\) −6.43000 + 6.43000i −0.252789 + 0.252789i −0.822113 0.569324i \(-0.807205\pi\)
0.569324 + 0.822113i \(0.307205\pi\)
\(648\) 0 0
\(649\) −7.88172 7.88172i −0.309384 0.309384i
\(650\) 0 0
\(651\) −5.86030 + 2.42742i −0.229683 + 0.0951380i
\(652\) 0 0
\(653\) 5.23125 12.6294i 0.204715 0.494225i −0.787861 0.615853i \(-0.788812\pi\)
0.992576 + 0.121628i \(0.0388115\pi\)
\(654\) 0 0
\(655\) 3.77811i 0.147623i
\(656\) 0 0
\(657\) 0.0601790i 0.00234781i
\(658\) 0 0
\(659\) −5.94335 + 14.3485i −0.231520 + 0.558939i −0.996357 0.0852857i \(-0.972820\pi\)
0.764837 + 0.644224i \(0.222820\pi\)
\(660\) 0 0
\(661\) 21.2370 8.79664i 0.826022 0.342150i 0.0706953 0.997498i \(-0.477478\pi\)
0.755327 + 0.655348i \(0.227478\pi\)
\(662\) 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
\(667\) 3.12534 + 7.54524i 0.121014 + 0.292153i
\(668\) 0 0
\(669\) −26.0811 10.8031i −1.00835 0.417673i
\(670\) 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
\(675\) −22.6187 9.36899i −0.870596 0.360613i
\(676\) 0 0
\(677\) −8.16911 19.7220i −0.313964 0.757977i −0.999550 0.0299855i \(-0.990454\pi\)
0.685586 0.727992i \(-0.259546\pi\)
\(678\) 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
\(683\) 22.6783 9.39365i 0.867760 0.359438i 0.0960227 0.995379i \(-0.469388\pi\)
0.771738 + 0.635941i \(0.219388\pi\)
\(684\) 0 0
\(685\) 1.64363 3.96808i 0.0627999 0.151612i
\(686\) 0 0
\(687\) 22.7192i 0.866793i
\(688\) 0 0
\(689\) 44.5614i 1.69766i
\(690\) 0 0
\(691\) −5.63891 + 13.6135i −0.214514 + 0.517883i −0.994107 0.108404i \(-0.965426\pi\)
0.779593 + 0.626287i \(0.215426\pi\)
\(692\) 0 0
\(693\) −0.206977 + 0.0857327i −0.00786241 + 0.00325671i
\(694\) 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
\(699\) −1.63341 3.94340i −0.0617812 0.149153i
\(700\) 0 0
\(701\) 42.5832 + 17.6385i 1.60835 + 0.666199i 0.992565 0.121718i \(-0.0388405\pi\)
0.615781 + 0.787917i \(0.288840\pi\)
\(702\) 0 0
\(703\) 16.1025 0.607316
\(704\) 0 0
\(705\) −9.31319 −0.350755
\(706\) 0 0
\(707\) 4.23483 + 1.75412i 0.159267 + 0.0659706i
\(708\) 0 0
\(709\) −11.1294 26.8688i −0.417974 1.00908i −0.982934 0.183960i \(-0.941108\pi\)
0.564960 0.825118i \(-0.308892\pi\)
\(710\) 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
\(715\) 11.6779 4.83714i 0.436728 0.180899i
\(716\) 0 0
\(717\) 8.42003 20.3277i 0.314452 0.759154i
\(718\) 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
\(723\) 12.5186 30.2226i 0.465573 1.12399i
\(724\) 0 0
\(725\) −5.80585 + 2.40486i −0.215624 + 0.0893144i
\(726\) 0 0
\(727\) 20.1795 + 20.1795i 0.748416 + 0.748416i 0.974182 0.225765i \(-0.0724882\pi\)
−0.225765 + 0.974182i \(0.572488\pi\)
\(728\) 0 0
\(729\) −19.2485 + 19.2485i −0.712909 + 0.712909i
\(730\) 0 0
\(731\) −1.11524 2.69243i −0.0412487 0.0995832i
\(732\) 0 0
\(733\) 11.2520 + 4.66073i 0.415602 + 0.172148i 0.580679 0.814133i \(-0.302787\pi\)
−0.165077 + 0.986281i \(0.552787\pi\)
\(734\) 0 0
\(735\) −1.17496 −0.0433391
\(736\) 0 0
\(737\) −21.5103 −0.792343
\(738\) 0 0
\(739\) −22.6586 9.38552i −0.833512 0.345252i −0.0752197 0.997167i \(-0.523966\pi\)
−0.758292 + 0.651915i \(0.773966\pi\)
\(740\) 0 0
\(741\) 16.6110 + 40.1024i 0.610219 + 1.47320i
\(742\) 0 0
\(743\) −30.4077 + 30.4077i −1.11555 + 1.11555i −0.123164 + 0.992386i \(0.539304\pi\)
−0.992386 + 0.123164i \(0.960696\pi\)
\(744\) 0 0
\(745\) 2.02215 + 2.02215i 0.0740859 + 0.0740859i
\(746\) 0 0
\(747\) −0.0217970 + 0.00902860i −0.000797510 + 0.000330339i
\(748\) 0 0
\(749\) −17.2944 + 41.7523i −0.631923 + 1.52560i
\(750\) 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
\(755\) 4.19124 10.1185i 0.152535 0.368251i
\(756\) 0 0
\(757\) 2.65558 1.09998i 0.0965185 0.0399793i −0.333902 0.942608i \(-0.608365\pi\)
0.430420 + 0.902629i \(0.358365\pi\)
\(758\) 0 0
\(759\) −27.7025 27.7025i −1.00554 1.00554i
\(760\) 0 0
\(761\) −17.9432 + 17.9432i −0.650442 + 0.650442i −0.953099 0.302658i \(-0.902126\pi\)
0.302658 + 0.953099i \(0.402126\pi\)
\(762\) 0 0
\(763\) 6.28528 + 15.1740i 0.227542 + 0.549336i
\(764\) 0 0
\(765\) 0.0675875 + 0.0279956i 0.00244363 + 0.00101218i
\(766\) 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
\(771\) −3.29077 1.36308i −0.118514 0.0490902i
\(772\) 0 0
\(773\) −7.20933 17.4049i −0.259302 0.626009i 0.739591 0.673056i \(-0.235019\pi\)
−0.998893 + 0.0470471i \(0.985019\pi\)
\(774\) 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
\(779\) −46.7225 + 19.3531i −1.67401 + 0.693396i
\(780\) 0 0
\(781\) −19.0561 + 46.0056i −0.681882 + 1.64621i
\(782\) 0 0
\(783\) 6.98779i 0.249723i
\(784\) 0 0
\(785\) 9.62716i 0.343608i
\(786\) 0 0
\(787\) 13.3060 32.1235i 0.474307 1.14508i −0.487934 0.872881i \(-0.662249\pi\)
0.962241 0.272198i \(-0.0877507\pi\)
\(788\) 0 0
\(789\) −40.2514 + 16.6727i −1.43299 + 0.593563i
\(790\) 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
\(795\) 2.66736 + 6.43957i 0.0946015 + 0.228388i
\(796\) 0 0
\(797\) −16.5225 6.84385i −0.585258 0.242422i 0.0703512 0.997522i \(-0.477588\pi\)
−0.655609 + 0.755101i \(0.727588\pi\)
\(798\) 0 0
\(799\) −51.2724 −1.81389
\(800\) 0 0
\(801\) −0.0406932 −0.00143782
\(802\) 0 0
\(803\) 8.27229 + 3.42649i 0.291923 + 0.120918i
\(804\) 0 0
\(805\) −3.11018 7.50864i −0.109620 0.264645i
\(806\) 0 0
\(807\) −18.0253 + 18.0253i −0.634521 + 0.634521i
\(808\) 0 0
\(809\) 10.5558 + 10.5558i 0.371123 + 0.371123i 0.867886 0.496763i \(-0.165478\pi\)
−0.496763 + 0.867886i \(0.665478\pi\)
\(810\) 0 0
\(811\) −4.31304 + 1.78652i −0.151451 + 0.0627333i −0.457122 0.889404i \(-0.651120\pi\)
0.305670 + 0.952138i \(0.401120\pi\)
\(812\) 0 0
\(813\) 1.58186 3.81894i 0.0554782 0.133936i
\(814\) 0 0
\(815\) 2.83121i 0.0991728i
\(816\) 0 0
\(817\) 2.27601i 0.0796274i
\(818\) 0 0
\(819\) −0.140797 + 0.339914i −0.00491985 + 0.0118776i
\(820\) 0 0
\(821\) −25.4218 + 10.5300i −0.887225 + 0.367501i −0.779295 0.626658i \(-0.784422\pi\)
−0.107931 + 0.994158i \(0.534422\pi\)
\(822\) 0 0
\(823\) 28.1280 + 28.1280i 0.980479 + 0.980479i 0.999813 0.0193343i \(-0.00615470\pi\)
−0.0193343 + 0.999813i \(0.506155\pi\)
\(824\) 0 0
\(825\) 21.3163 21.3163i 0.742138 0.742138i
\(826\) 0 0
\(827\) −6.05658 14.6219i −0.210608 0.508453i 0.782909 0.622136i \(-0.213735\pi\)
−0.993517 + 0.113683i \(0.963735\pi\)
\(828\) 0 0
\(829\) −28.3516 11.7436i −0.984691 0.407872i −0.168530 0.985697i \(-0.553902\pi\)
−0.816161 + 0.577824i \(0.803902\pi\)
\(830\) 0 0
\(831\) 17.2507 0.598419
\(832\) 0 0
\(833\) −6.46858 −0.224123
\(834\) 0 0
\(835\) −2.55671 1.05902i −0.0884786 0.0366490i
\(836\) 0 0
\(837\) 3.05640 + 7.37881i 0.105645 + 0.255049i
\(838\) 0 0
\(839\) 19.2678 19.2678i 0.665199 0.665199i −0.291401 0.956601i \(-0.594122\pi\)
0.956601 + 0.291401i \(0.0941215\pi\)
\(840\) 0 0
\(841\) −19.2378 19.2378i −0.663372 0.663372i
\(842\) 0 0
\(843\) −44.7710 + 18.5448i −1.54199 + 0.638715i
\(844\) 0 0
\(845\) 5.18414 12.5156i 0.178340 0.430551i
\(846\) 0 0
\(847\) 6.90528i 0.237268i
\(848\) 0 0
\(849\) 8.89609i 0.305313i
\(850\) 0 0
\(851\) −9.13394 + 22.0513i −0.313107 + 0.755908i
\(852\) 0 0
\(853\) −13.3619 + 5.53470i −0.457504 + 0.189504i −0.599520 0.800360i \(-0.704642\pi\)
0.142016 + 0.989864i \(0.454642\pi\)
\(854\) 0 0
\(855\) 0.0403999 + 0.0403999i 0.00138165 + 0.00138165i
\(856\) 0 0
\(857\) 16.2115 16.2115i 0.553775 0.553775i −0.373753 0.927528i \(-0.621929\pi\)
0.927528 + 0.373753i \(0.121929\pi\)
\(858\) 0 0
\(859\) 10.6184 + 25.6351i 0.362295 + 0.874658i 0.994964 + 0.100235i \(0.0319595\pi\)
−0.632669 + 0.774423i \(0.718040\pi\)
\(860\) 0 0
\(861\) 47.0617 + 19.4936i 1.60386 + 0.664341i
\(862\) 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
\(867\) −17.1277 7.09454i −0.581689 0.240943i
\(868\) 0 0
\(869\) 18.2927 + 44.1625i 0.620538 + 1.49811i
\(870\) 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
\(875\) 11.9343 4.94336i 0.403454 0.167116i
\(876\) 0 0
\(877\) −18.1786 + 43.8869i −0.613846 + 1.48196i 0.244896 + 0.969549i \(0.421246\pi\)
−0.858743 + 0.512407i \(0.828754\pi\)
\(878\) 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
\(883\) 20.6785 49.9223i 0.695886 1.68002i −0.0366838 0.999327i \(-0.511679\pi\)
0.732570 0.680692i \(-0.238321\pi\)
\(884\) 0 0
\(885\) 2.64534 1.09574i 0.0889221 0.0368327i
\(886\) 0 0
\(887\) −17.8530 17.8530i −0.599446 0.599446i 0.340719 0.940165i \(-0.389329\pi\)
−0.940165 + 0.340719i \(0.889329\pi\)
\(888\) 0 0
\(889\) −7.42029 + 7.42029i −0.248868 + 0.248868i
\(890\) 0 0
\(891\) −12.7209 30.7109i −0.426165 1.02885i
\(892\) 0 0
\(893\) −36.9950 15.3238i −1.23799 0.512792i
\(894\) 0 0
\(895\) 12.5531 0.419604
\(896\) 0 0
\(897\) −64.3400 −2.14825
\(898\) 0 0
\(899\) 1.89402 + 0.784527i 0.0631690 + 0.0261655i
\(900\) 0 0
\(901\) 14.6847 + 35.4521i 0.489220 + 1.18108i
\(902\) 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
\(907\) −10.2527 + 4.24679i −0.340434 + 0.141012i −0.546349 0.837558i \(-0.683983\pi\)
0.205915 + 0.978570i \(0.433983\pi\)
\(908\) 0 0
\(909\) 0.0182783 0.0441276i 0.000606252 0.00146362i
\(910\) 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
\(915\) −2.93340 + 7.08185i −0.0969752 + 0.234119i
\(916\) 0 0
\(917\) 15.1168 6.26160i 0.499202 0.206776i
\(918\) 0 0
\(919\) −5.17031 5.17031i −0.170553 0.170553i 0.616669 0.787222i \(-0.288482\pi\)
−0.787222 + 0.616669i \(0.788482\pi\)
\(920\) 0 0
\(921\) −26.3169 + 26.3169i −0.867172 + 0.867172i
\(922\) 0 0
\(923\) 31.2955 + 75.5541i 1.03011 + 2.48689i
\(924\) 0 0
\(925\) −16.9678 7.02831i −0.557899 0.231089i
\(926\) 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
\(931\) −4.66733 1.93327i −0.152965 0.0633604i
\(932\) 0 0
\(933\) −14.4498 34.8848i −0.473064 1.14208i
\(934\) 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 \(-0.981612\pi\)
−0.0577344 0.998332i \(-0.518388\pi\)
\(938\) 0 0
\(939\) 31.7569 13.1541i 1.03635 0.429269i
\(940\) 0 0
\(941\) −9.90288 + 23.9077i −0.322825 + 0.779368i 0.676263 + 0.736660i \(0.263598\pi\)
−0.999088 + 0.0427073i \(0.986402\pi\)
\(942\) 0 0
\(943\) 74.9612i 2.44107i
\(944\) 0 0
\(945\) 6.95390i 0.226210i
\(946\) 0 0
\(947\) 8.97880 21.6767i 0.291772 0.704399i −0.708227 0.705985i \(-0.750505\pi\)
0.999999 + 0.00158544i \(0.000504662\pi\)
\(948\) 0 0
\(949\) 13.5854 5.62726i 0.441001 0.182669i
\(950\) 0 0
\(951\) 9.74220 + 9.74220i 0.315912 + 0.315912i
\(952\) 0 0
\(953\) −3.00440 + 3.00440i −0.0973222 + 0.0973222i −0.754092 0.656769i \(-0.771923\pi\)
0.656769 + 0.754092i \(0.271923\pi\)
\(954\) 0 0
\(955\) 2.21561 + 5.34895i 0.0716954 + 0.173088i
\(956\) 0 0
\(957\) −7.94930 3.29271i −0.256965 0.106438i
\(958\) 0 0
\(959\) −18.6010 −0.600657
\(960\) 0 0
\(961\) −28.6569 −0.924415
\(962\) 0 0
\(963\) 0.435066 + 0.180210i 0.0140198 + 0.00580720i
\(964\) 0 0
\(965\) −0.810542 1.95682i −0.0260923 0.0629923i
\(966\) 0 0
\(967\) 29.1087 29.1087i 0.936074 0.936074i −0.0620023 0.998076i \(-0.519749\pi\)
0.998076 + 0.0620023i \(0.0197486\pi\)
\(968\) 0 0
\(969\) −26.4307 26.4307i −0.849075 0.849075i
\(970\) 0 0
\(971\) −0.623957 + 0.258452i −0.0200237 + 0.00829411i −0.392673 0.919678i \(-0.628450\pi\)
0.372649 + 0.927972i \(0.378450\pi\)
\(972\) 0 0
\(973\) 10.6739 25.7690i 0.342189 0.826117i
\(974\) 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
\(979\) 2.31700 5.59374i 0.0740518 0.178777i
\(980\) 0 0
\(981\) 0.158116 0.0654936i 0.00504824 0.00209105i
\(982\) 0 0
\(983\) 1.49880 + 1.49880i 0.0478044 + 0.0478044i 0.730605 0.682800i \(-0.239238\pi\)
−0.682800 + 0.730605i \(0.739238\pi\)
\(984\) 0 0
\(985\) −6.23264 + 6.23264i −0.198588 + 0.198588i
\(986\) 0 0
\(987\) 15.4351 + 37.2636i 0.491304 + 1.18611i
\(988\) 0 0
\(989\) 3.11684 + 1.29104i 0.0991098 + 0.0410526i
\(990\) 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
\(995\) −11.6146 4.81094i −0.368209 0.152517i
\(996\) 0 0
\(997\) 2.47449 + 5.97395i 0.0783679 + 0.189197i 0.958208 0.286074i \(-0.0923502\pi\)
−0.879840 + 0.475270i \(0.842350\pi\)
\(998\) 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.c.641.3 yes 16
4.3 odd 2 inner 1024.2.g.c.641.2 yes 16
8.3 odd 2 1024.2.g.h.641.3 yes 16
8.5 even 2 1024.2.g.h.641.2 yes 16
16.3 odd 4 1024.2.g.b.129.3 yes 16
16.5 even 4 1024.2.g.e.129.3 yes 16
16.11 odd 4 1024.2.g.e.129.2 yes 16
16.13 even 4 1024.2.g.b.129.2 16
32.3 odd 8 1024.2.g.e.897.2 yes 16
32.5 even 8 1024.2.g.h.385.2 yes 16
32.11 odd 8 inner 1024.2.g.c.385.2 yes 16
32.13 even 8 1024.2.g.b.897.2 yes 16
32.19 odd 8 1024.2.g.b.897.3 yes 16
32.21 even 8 inner 1024.2.g.c.385.3 yes 16
32.27 odd 8 1024.2.g.h.385.3 yes 16
32.29 even 8 1024.2.g.e.897.3 yes 16
64.11 odd 16 4096.2.a.o.1.3 8
64.21 even 16 4096.2.a.o.1.4 8
64.43 odd 16 4096.2.a.n.1.6 8
64.53 even 16 4096.2.a.n.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1024.2.g.b.129.2 16 16.13 even 4
1024.2.g.b.129.3 yes 16 16.3 odd 4
1024.2.g.b.897.2 yes 16 32.13 even 8
1024.2.g.b.897.3 yes 16 32.19 odd 8
1024.2.g.c.385.2 yes 16 32.11 odd 8 inner
1024.2.g.c.385.3 yes 16 32.21 even 8 inner
1024.2.g.c.641.2 yes 16 4.3 odd 2 inner
1024.2.g.c.641.3 yes 16 1.1 even 1 trivial
1024.2.g.e.129.2 yes 16 16.11 odd 4
1024.2.g.e.129.3 yes 16 16.5 even 4
1024.2.g.e.897.2 yes 16 32.3 odd 8
1024.2.g.e.897.3 yes 16 32.29 even 8
1024.2.g.h.385.2 yes 16 32.5 even 8
1024.2.g.h.385.3 yes 16 32.27 odd 8
1024.2.g.h.641.2 yes 16 8.5 even 2
1024.2.g.h.641.3 yes 16 8.3 odd 2
4096.2.a.n.1.5 8 64.53 even 16
4096.2.a.n.1.6 8 64.43 odd 16
4096.2.a.o.1.3 8 64.11 odd 16
4096.2.a.o.1.4 8 64.21 even 16