Properties

Label 124.2.f.b.109.1
Level $124$
Weight $2$
Character 124.109
Analytic conductor $0.990$
Analytic rank $0$
Dimension $4$
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] = [124,2,Mod(33,124)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(124, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("124.33");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 124.f (of order \(5\), 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: \(0.990144985064\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 109.1
Root \(0.809017 - 0.587785i\) of defining polynomial
Character \(\chi\) \(=\) 124.109
Dual form 124.2.f.b.33.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.80902 + 1.31433i) q^{3} -0.381966 q^{5} +(0.0729490 - 0.224514i) q^{7} +(0.618034 + 1.90211i) q^{9} +O(q^{10})\) \(q+(1.80902 + 1.31433i) q^{3} -0.381966 q^{5} +(0.0729490 - 0.224514i) q^{7} +(0.618034 + 1.90211i) q^{9} +(-0.381966 + 1.17557i) q^{11} +(-3.73607 - 2.71441i) q^{13} +(-0.690983 - 0.502029i) q^{15} +(0.690983 + 2.12663i) q^{17} +(5.42705 - 3.94298i) q^{19} +(0.427051 - 0.310271i) q^{21} +(-0.545085 - 1.67760i) q^{23} -4.85410 q^{25} +(0.690983 - 2.12663i) q^{27} +(-4.54508 + 3.30220i) q^{29} +(-4.19098 - 3.66547i) q^{31} +(-2.23607 + 1.62460i) q^{33} +(-0.0278640 + 0.0857567i) q^{35} +4.70820 q^{37} +(-3.19098 - 9.82084i) q^{39} +(-8.47214 + 6.15537i) q^{41} +(-0.927051 + 0.673542i) q^{43} +(-0.236068 - 0.726543i) q^{45} +(6.92705 + 5.03280i) q^{47} +(5.61803 + 4.08174i) q^{49} +(-1.54508 + 4.75528i) q^{51} +(-2.83688 - 8.73102i) q^{53} +(0.145898 - 0.449028i) q^{55} +15.0000 q^{57} +(10.2812 + 7.46969i) q^{59} +2.00000 q^{61} +0.472136 q^{63} +(1.42705 + 1.03681i) q^{65} +3.00000 q^{67} +(1.21885 - 3.75123i) q^{69} +(4.04508 + 12.4495i) q^{71} +(0.0450850 - 0.138757i) q^{73} +(-8.78115 - 6.37988i) q^{75} +(0.236068 + 0.171513i) q^{77} +(8.89919 - 6.46564i) q^{81} +(-6.35410 + 4.61653i) q^{83} +(-0.263932 - 0.812299i) q^{85} -12.5623 q^{87} +(-4.95492 + 15.2497i) q^{89} +(-0.881966 + 0.640786i) q^{91} +(-2.76393 - 12.1392i) q^{93} +(-2.07295 + 1.50609i) q^{95} +(1.45492 - 4.47777i) q^{97} -2.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 5 q^{3} - 6 q^{5} + 7 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 5 q^{3} - 6 q^{5} + 7 q^{7} - 2 q^{9} - 6 q^{11} - 6 q^{13} - 5 q^{15} + 5 q^{17} + 15 q^{19} - 5 q^{21} + 9 q^{23} - 6 q^{25} + 5 q^{27} - 7 q^{29} - 19 q^{31} - 18 q^{35} - 8 q^{37} - 15 q^{39} - 16 q^{41} + 3 q^{43} + 8 q^{45} + 21 q^{47} + 18 q^{49} + 5 q^{51} - 27 q^{53} + 14 q^{55} + 60 q^{57} + 21 q^{59} + 8 q^{61} - 16 q^{63} - q^{65} + 12 q^{67} + 25 q^{69} + 5 q^{71} - 11 q^{73} - 15 q^{75} - 8 q^{77} + 11 q^{81} - 12 q^{83} - 10 q^{85} - 10 q^{87} - 31 q^{89} - 8 q^{91} - 20 q^{93} - 15 q^{95} + 17 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/124\mathbb{Z}\right)^\times\).

\(n\) \(63\) \(65\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{5}\right)\)

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
\(3\) 1.80902 + 1.31433i 1.04444 + 0.758827i 0.971147 0.238483i \(-0.0766502\pi\)
0.0732898 + 0.997311i \(0.476650\pi\)
\(4\) 0 0
\(5\) −0.381966 −0.170820 −0.0854102 0.996346i \(-0.527220\pi\)
−0.0854102 + 0.996346i \(0.527220\pi\)
\(6\) 0 0
\(7\) 0.0729490 0.224514i 0.0275721 0.0848583i −0.936324 0.351138i \(-0.885795\pi\)
0.963896 + 0.266280i \(0.0857946\pi\)
\(8\) 0 0
\(9\) 0.618034 + 1.90211i 0.206011 + 0.634038i
\(10\) 0 0
\(11\) −0.381966 + 1.17557i −0.115167 + 0.354448i −0.991982 0.126380i \(-0.959664\pi\)
0.876815 + 0.480828i \(0.159664\pi\)
\(12\) 0 0
\(13\) −3.73607 2.71441i −1.03620 0.752843i −0.0666589 0.997776i \(-0.521234\pi\)
−0.969540 + 0.244933i \(0.921234\pi\)
\(14\) 0 0
\(15\) −0.690983 0.502029i −0.178411 0.129623i
\(16\) 0 0
\(17\) 0.690983 + 2.12663i 0.167588 + 0.515783i 0.999218 0.0395478i \(-0.0125917\pi\)
−0.831630 + 0.555331i \(0.812592\pi\)
\(18\) 0 0
\(19\) 5.42705 3.94298i 1.24505 0.904582i 0.247127 0.968983i \(-0.420514\pi\)
0.997924 + 0.0644007i \(0.0205136\pi\)
\(20\) 0 0
\(21\) 0.427051 0.310271i 0.0931902 0.0677066i
\(22\) 0 0
\(23\) −0.545085 1.67760i −0.113658 0.349804i 0.878007 0.478648i \(-0.158873\pi\)
−0.991665 + 0.128845i \(0.958873\pi\)
\(24\) 0 0
\(25\) −4.85410 −0.970820
\(26\) 0 0
\(27\) 0.690983 2.12663i 0.132980 0.409270i
\(28\) 0 0
\(29\) −4.54508 + 3.30220i −0.844001 + 0.613203i −0.923485 0.383633i \(-0.874673\pi\)
0.0794844 + 0.996836i \(0.474673\pi\)
\(30\) 0 0
\(31\) −4.19098 3.66547i −0.752723 0.658338i
\(32\) 0 0
\(33\) −2.23607 + 1.62460i −0.389249 + 0.282806i
\(34\) 0 0
\(35\) −0.0278640 + 0.0857567i −0.00470988 + 0.0144955i
\(36\) 0 0
\(37\) 4.70820 0.774024 0.387012 0.922075i \(-0.373507\pi\)
0.387012 + 0.922075i \(0.373507\pi\)
\(38\) 0 0
\(39\) −3.19098 9.82084i −0.510966 1.57259i
\(40\) 0 0
\(41\) −8.47214 + 6.15537i −1.32313 + 0.961307i −0.323238 + 0.946318i \(0.604771\pi\)
−0.999888 + 0.0149890i \(0.995229\pi\)
\(42\) 0 0
\(43\) −0.927051 + 0.673542i −0.141374 + 0.102714i −0.656224 0.754566i \(-0.727847\pi\)
0.514850 + 0.857280i \(0.327847\pi\)
\(44\) 0 0
\(45\) −0.236068 0.726543i −0.0351909 0.108307i
\(46\) 0 0
\(47\) 6.92705 + 5.03280i 1.01041 + 0.734109i 0.964296 0.264827i \(-0.0853149\pi\)
0.0461183 + 0.998936i \(0.485315\pi\)
\(48\) 0 0
\(49\) 5.61803 + 4.08174i 0.802576 + 0.583106i
\(50\) 0 0
\(51\) −1.54508 + 4.75528i −0.216355 + 0.665873i
\(52\) 0 0
\(53\) −2.83688 8.73102i −0.389676 1.19930i −0.933031 0.359795i \(-0.882846\pi\)
0.543356 0.839503i \(-0.317154\pi\)
\(54\) 0 0
\(55\) 0.145898 0.449028i 0.0196729 0.0605469i
\(56\) 0 0
\(57\) 15.0000 1.98680
\(58\) 0 0
\(59\) 10.2812 + 7.46969i 1.33849 + 0.972471i 0.999498 + 0.0316830i \(0.0100867\pi\)
0.338994 + 0.940788i \(0.389913\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 0.472136 0.0594835
\(64\) 0 0
\(65\) 1.42705 + 1.03681i 0.177004 + 0.128601i
\(66\) 0 0
\(67\) 3.00000 0.366508 0.183254 0.983066i \(-0.441337\pi\)
0.183254 + 0.983066i \(0.441337\pi\)
\(68\) 0 0
\(69\) 1.21885 3.75123i 0.146732 0.451594i
\(70\) 0 0
\(71\) 4.04508 + 12.4495i 0.480063 + 1.47748i 0.839005 + 0.544123i \(0.183138\pi\)
−0.358942 + 0.933360i \(0.616862\pi\)
\(72\) 0 0
\(73\) 0.0450850 0.138757i 0.00527680 0.0162403i −0.948383 0.317126i \(-0.897282\pi\)
0.953660 + 0.300886i \(0.0972822\pi\)
\(74\) 0 0
\(75\) −8.78115 6.37988i −1.01396 0.736685i
\(76\) 0 0
\(77\) 0.236068 + 0.171513i 0.0269024 + 0.0195458i
\(78\) 0 0
\(79\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(80\) 0 0
\(81\) 8.89919 6.46564i 0.988799 0.718404i
\(82\) 0 0
\(83\) −6.35410 + 4.61653i −0.697453 + 0.506729i −0.879102 0.476634i \(-0.841857\pi\)
0.181649 + 0.983364i \(0.441857\pi\)
\(84\) 0 0
\(85\) −0.263932 0.812299i −0.0286274 0.0881062i
\(86\) 0 0
\(87\) −12.5623 −1.34682
\(88\) 0 0
\(89\) −4.95492 + 15.2497i −0.525220 + 1.61646i 0.238661 + 0.971103i \(0.423292\pi\)
−0.763880 + 0.645358i \(0.776708\pi\)
\(90\) 0 0
\(91\) −0.881966 + 0.640786i −0.0924552 + 0.0671726i
\(92\) 0 0
\(93\) −2.76393 12.1392i −0.286606 1.25878i
\(94\) 0 0
\(95\) −2.07295 + 1.50609i −0.212680 + 0.154521i
\(96\) 0 0
\(97\) 1.45492 4.47777i 0.147724 0.454648i −0.849627 0.527384i \(-0.823173\pi\)
0.997351 + 0.0727356i \(0.0231729\pi\)
\(98\) 0 0
\(99\) −2.47214 −0.248459
\(100\) 0 0
\(101\) −2.85410 8.78402i −0.283994 0.874043i −0.986698 0.162561i \(-0.948024\pi\)
0.702705 0.711482i \(-0.251976\pi\)
\(102\) 0 0
\(103\) −5.78115 + 4.20025i −0.569634 + 0.413863i −0.834972 0.550292i \(-0.814516\pi\)
0.265338 + 0.964155i \(0.414516\pi\)
\(104\) 0 0
\(105\) −0.163119 + 0.118513i −0.0159188 + 0.0115657i
\(106\) 0 0
\(107\) −1.26393 3.88998i −0.122189 0.376059i 0.871190 0.490947i \(-0.163349\pi\)
−0.993378 + 0.114888i \(0.963349\pi\)
\(108\) 0 0
\(109\) −5.66312 4.11450i −0.542428 0.394097i 0.282558 0.959250i \(-0.408817\pi\)
−0.824986 + 0.565153i \(0.808817\pi\)
\(110\) 0 0
\(111\) 8.51722 + 6.18812i 0.808419 + 0.587351i
\(112\) 0 0
\(113\) 2.44427 7.52270i 0.229938 0.707676i −0.767815 0.640672i \(-0.778656\pi\)
0.997753 0.0670040i \(-0.0213440\pi\)
\(114\) 0 0
\(115\) 0.208204 + 0.640786i 0.0194151 + 0.0597536i
\(116\) 0 0
\(117\) 2.85410 8.78402i 0.263862 0.812083i
\(118\) 0 0
\(119\) 0.527864 0.0483892
\(120\) 0 0
\(121\) 7.66312 + 5.56758i 0.696647 + 0.506144i
\(122\) 0 0
\(123\) −23.4164 −2.11139
\(124\) 0 0
\(125\) 3.76393 0.336656
\(126\) 0 0
\(127\) 7.28115 + 5.29007i 0.646098 + 0.469418i 0.861940 0.507011i \(-0.169250\pi\)
−0.215842 + 0.976428i \(0.569250\pi\)
\(128\) 0 0
\(129\) −2.56231 −0.225598
\(130\) 0 0
\(131\) 4.33688 13.3475i 0.378915 1.16618i −0.561884 0.827216i \(-0.689923\pi\)
0.940799 0.338965i \(-0.110077\pi\)
\(132\) 0 0
\(133\) −0.489357 1.50609i −0.0424326 0.130594i
\(134\) 0 0
\(135\) −0.263932 + 0.812299i −0.0227157 + 0.0699116i
\(136\) 0 0
\(137\) −8.47214 6.15537i −0.723823 0.525888i 0.163780 0.986497i \(-0.447631\pi\)
−0.887603 + 0.460608i \(0.847631\pi\)
\(138\) 0 0
\(139\) −11.3992 8.28199i −0.966866 0.702470i −0.0121312 0.999926i \(-0.503862\pi\)
−0.954735 + 0.297457i \(0.903862\pi\)
\(140\) 0 0
\(141\) 5.91641 + 18.2088i 0.498251 + 1.53346i
\(142\) 0 0
\(143\) 4.61803 3.35520i 0.386179 0.280576i
\(144\) 0 0
\(145\) 1.73607 1.26133i 0.144173 0.104748i
\(146\) 0 0
\(147\) 4.79837 + 14.7679i 0.395763 + 1.21803i
\(148\) 0 0
\(149\) 5.56231 0.455682 0.227841 0.973698i \(-0.426833\pi\)
0.227841 + 0.973698i \(0.426833\pi\)
\(150\) 0 0
\(151\) 1.66312 5.11855i 0.135343 0.416542i −0.860300 0.509787i \(-0.829724\pi\)
0.995643 + 0.0932453i \(0.0297241\pi\)
\(152\) 0 0
\(153\) −3.61803 + 2.62866i −0.292501 + 0.212514i
\(154\) 0 0
\(155\) 1.60081 + 1.40008i 0.128580 + 0.112457i
\(156\) 0 0
\(157\) −15.9443 + 11.5842i −1.27249 + 0.924519i −0.999299 0.0374402i \(-0.988080\pi\)
−0.273193 + 0.961959i \(0.588080\pi\)
\(158\) 0 0
\(159\) 6.34346 19.5232i 0.503069 1.54829i
\(160\) 0 0
\(161\) −0.416408 −0.0328175
\(162\) 0 0
\(163\) −2.45492 7.55545i −0.192284 0.591789i −0.999998 0.00221381i \(-0.999295\pi\)
0.807714 0.589575i \(-0.200705\pi\)
\(164\) 0 0
\(165\) 0.854102 0.620541i 0.0664917 0.0483091i
\(166\) 0 0
\(167\) −15.0902 + 10.9637i −1.16771 + 0.848393i −0.990733 0.135822i \(-0.956633\pi\)
−0.176979 + 0.984215i \(0.556633\pi\)
\(168\) 0 0
\(169\) 2.57295 + 7.91872i 0.197919 + 0.609133i
\(170\) 0 0
\(171\) 10.8541 + 7.88597i 0.830034 + 0.603055i
\(172\) 0 0
\(173\) −7.78115 5.65334i −0.591590 0.429815i 0.251294 0.967911i \(-0.419144\pi\)
−0.842884 + 0.538095i \(0.819144\pi\)
\(174\) 0 0
\(175\) −0.354102 + 1.08981i −0.0267676 + 0.0823822i
\(176\) 0 0
\(177\) 8.78115 + 27.0256i 0.660032 + 2.03137i
\(178\) 0 0
\(179\) 2.28115 7.02067i 0.170501 0.524749i −0.828898 0.559400i \(-0.811032\pi\)
0.999399 + 0.0346503i \(0.0110317\pi\)
\(180\) 0 0
\(181\) 16.4164 1.22022 0.610111 0.792316i \(-0.291125\pi\)
0.610111 + 0.792316i \(0.291125\pi\)
\(182\) 0 0
\(183\) 3.61803 + 2.62866i 0.267453 + 0.194316i
\(184\) 0 0
\(185\) −1.79837 −0.132219
\(186\) 0 0
\(187\) −2.76393 −0.202119
\(188\) 0 0
\(189\) −0.427051 0.310271i −0.0310634 0.0225689i
\(190\) 0 0
\(191\) 21.3820 1.54714 0.773572 0.633708i \(-0.218468\pi\)
0.773572 + 0.633708i \(0.218468\pi\)
\(192\) 0 0
\(193\) 3.51722 10.8249i 0.253175 0.779193i −0.741009 0.671496i \(-0.765652\pi\)
0.994184 0.107697i \(-0.0343477\pi\)
\(194\) 0 0
\(195\) 1.21885 + 3.75123i 0.0872835 + 0.268631i
\(196\) 0 0
\(197\) 8.30902 25.5725i 0.591993 1.82197i 0.0228342 0.999739i \(-0.492731\pi\)
0.569159 0.822228i \(-0.307269\pi\)
\(198\) 0 0
\(199\) 17.7533 + 12.8985i 1.25850 + 0.914352i 0.998683 0.0513068i \(-0.0163386\pi\)
0.259814 + 0.965659i \(0.416339\pi\)
\(200\) 0 0
\(201\) 5.42705 + 3.94298i 0.382795 + 0.278117i
\(202\) 0 0
\(203\) 0.409830 + 1.26133i 0.0287644 + 0.0885278i
\(204\) 0 0
\(205\) 3.23607 2.35114i 0.226017 0.164211i
\(206\) 0 0
\(207\) 2.85410 2.07363i 0.198374 0.144127i
\(208\) 0 0
\(209\) 2.56231 + 7.88597i 0.177238 + 0.545484i
\(210\) 0 0
\(211\) −9.88854 −0.680755 −0.340378 0.940289i \(-0.610555\pi\)
−0.340378 + 0.940289i \(0.610555\pi\)
\(212\) 0 0
\(213\) −9.04508 + 27.8379i −0.619759 + 1.90742i
\(214\) 0 0
\(215\) 0.354102 0.257270i 0.0241496 0.0175457i
\(216\) 0 0
\(217\) −1.12868 + 0.673542i −0.0766196 + 0.0457230i
\(218\) 0 0
\(219\) 0.263932 0.191758i 0.0178349 0.0129578i
\(220\) 0 0
\(221\) 3.19098 9.82084i 0.214649 0.660621i
\(222\) 0 0
\(223\) −3.47214 −0.232511 −0.116256 0.993219i \(-0.537089\pi\)
−0.116256 + 0.993219i \(0.537089\pi\)
\(224\) 0 0
\(225\) −3.00000 9.23305i −0.200000 0.615537i
\(226\) 0 0
\(227\) 19.6803 14.2986i 1.30623 0.949032i 0.306234 0.951956i \(-0.400931\pi\)
0.999996 + 0.00292450i \(0.000930897\pi\)
\(228\) 0 0
\(229\) −16.7082 + 12.1392i −1.10411 + 0.802182i −0.981726 0.190301i \(-0.939054\pi\)
−0.122383 + 0.992483i \(0.539054\pi\)
\(230\) 0 0
\(231\) 0.201626 + 0.620541i 0.0132660 + 0.0408286i
\(232\) 0 0
\(233\) −16.2533 11.8087i −1.06479 0.773614i −0.0898201 0.995958i \(-0.528629\pi\)
−0.974968 + 0.222344i \(0.928629\pi\)
\(234\) 0 0
\(235\) −2.64590 1.92236i −0.172599 0.125401i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −0.909830 2.80017i −0.0588520 0.181128i 0.917309 0.398177i \(-0.130357\pi\)
−0.976161 + 0.217049i \(0.930357\pi\)
\(240\) 0 0
\(241\) −2.92705 + 9.00854i −0.188548 + 0.580291i −0.999991 0.00414002i \(-0.998682\pi\)
0.811444 + 0.584431i \(0.198682\pi\)
\(242\) 0 0
\(243\) 17.8885 1.14755
\(244\) 0 0
\(245\) −2.14590 1.55909i −0.137096 0.0996064i
\(246\) 0 0
\(247\) −30.9787 −1.97113
\(248\) 0 0
\(249\) −17.5623 −1.11297
\(250\) 0 0
\(251\) 0.763932 + 0.555029i 0.0482190 + 0.0350331i 0.611634 0.791141i \(-0.290513\pi\)
−0.563415 + 0.826174i \(0.690513\pi\)
\(252\) 0 0
\(253\) 2.18034 0.137077
\(254\) 0 0
\(255\) 0.590170 1.81636i 0.0369579 0.113745i
\(256\) 0 0
\(257\) −7.85410 24.1724i −0.489925 1.50784i −0.824718 0.565544i \(-0.808666\pi\)
0.334793 0.942292i \(-0.391334\pi\)
\(258\) 0 0
\(259\) 0.343459 1.05706i 0.0213415 0.0656824i
\(260\) 0 0
\(261\) −9.09017 6.60440i −0.562667 0.408802i
\(262\) 0 0
\(263\) −23.2533 16.8945i −1.43386 1.04176i −0.989282 0.146019i \(-0.953354\pi\)
−0.444577 0.895740i \(-0.646646\pi\)
\(264\) 0 0
\(265\) 1.08359 + 3.33495i 0.0665645 + 0.204865i
\(266\) 0 0
\(267\) −29.0066 + 21.0745i −1.77517 + 1.28974i
\(268\) 0 0
\(269\) −10.0172 + 7.27794i −0.610761 + 0.443744i −0.849682 0.527295i \(-0.823206\pi\)
0.238921 + 0.971039i \(0.423206\pi\)
\(270\) 0 0
\(271\) 5.82624 + 17.9313i 0.353919 + 1.08925i 0.956634 + 0.291293i \(0.0940856\pi\)
−0.602715 + 0.797957i \(0.705914\pi\)
\(272\) 0 0
\(273\) −2.43769 −0.147536
\(274\) 0 0
\(275\) 1.85410 5.70634i 0.111807 0.344105i
\(276\) 0 0
\(277\) 10.8262 7.86572i 0.650486 0.472605i −0.212951 0.977063i \(-0.568307\pi\)
0.863437 + 0.504457i \(0.168307\pi\)
\(278\) 0 0
\(279\) 4.38197 10.2371i 0.262341 0.612880i
\(280\) 0 0
\(281\) 3.35410 2.43690i 0.200089 0.145373i −0.483229 0.875494i \(-0.660536\pi\)
0.683318 + 0.730121i \(0.260536\pi\)
\(282\) 0 0
\(283\) 5.86475 18.0498i 0.348623 1.07295i −0.610993 0.791636i \(-0.709230\pi\)
0.959616 0.281315i \(-0.0907705\pi\)
\(284\) 0 0
\(285\) −5.72949 −0.339386
\(286\) 0 0
\(287\) 0.763932 + 2.35114i 0.0450935 + 0.138783i
\(288\) 0 0
\(289\) 9.70820 7.05342i 0.571071 0.414907i
\(290\) 0 0
\(291\) 8.51722 6.18812i 0.499288 0.362754i
\(292\) 0 0
\(293\) 8.36475 + 25.7440i 0.488674 + 1.50398i 0.826589 + 0.562806i \(0.190278\pi\)
−0.337915 + 0.941177i \(0.609722\pi\)
\(294\) 0 0
\(295\) −3.92705 2.85317i −0.228642 0.166118i
\(296\) 0 0
\(297\) 2.23607 + 1.62460i 0.129750 + 0.0942688i
\(298\) 0 0
\(299\) −2.51722 + 7.74721i −0.145575 + 0.448033i
\(300\) 0 0
\(301\) 0.0835921 + 0.257270i 0.00481817 + 0.0148288i
\(302\) 0 0
\(303\) 6.38197 19.6417i 0.366634 1.12838i
\(304\) 0 0
\(305\) −0.763932 −0.0437426
\(306\) 0 0
\(307\) 20.9164 + 15.1967i 1.19376 + 0.867319i 0.993657 0.112455i \(-0.0358715\pi\)
0.200106 + 0.979774i \(0.435871\pi\)
\(308\) 0 0
\(309\) −15.9787 −0.908997
\(310\) 0 0
\(311\) 18.9443 1.07423 0.537116 0.843509i \(-0.319514\pi\)
0.537116 + 0.843509i \(0.319514\pi\)
\(312\) 0 0
\(313\) −11.5623 8.40051i −0.653540 0.474825i 0.210935 0.977500i \(-0.432349\pi\)
−0.864475 + 0.502675i \(0.832349\pi\)
\(314\) 0 0
\(315\) −0.180340 −0.0101610
\(316\) 0 0
\(317\) 1.52786 4.70228i 0.0858134 0.264106i −0.898937 0.438077i \(-0.855660\pi\)
0.984751 + 0.173970i \(0.0556597\pi\)
\(318\) 0 0
\(319\) −2.14590 6.60440i −0.120147 0.369775i
\(320\) 0 0
\(321\) 2.82624 8.69827i 0.157745 0.485490i
\(322\) 0 0
\(323\) 12.1353 + 8.81678i 0.675224 + 0.490579i
\(324\) 0 0
\(325\) 18.1353 + 13.1760i 1.00596 + 0.730875i
\(326\) 0 0
\(327\) −4.83688 14.8864i −0.267480 0.823219i
\(328\) 0 0
\(329\) 1.63525 1.18808i 0.0901545 0.0655011i
\(330\) 0 0
\(331\) −12.5451 + 9.11454i −0.689540 + 0.500980i −0.876509 0.481386i \(-0.840134\pi\)
0.186969 + 0.982366i \(0.440134\pi\)
\(332\) 0 0
\(333\) 2.90983 + 8.95554i 0.159458 + 0.490760i
\(334\) 0 0
\(335\) −1.14590 −0.0626071
\(336\) 0 0
\(337\) −10.3541 + 31.8666i −0.564024 + 1.73589i 0.106810 + 0.994279i \(0.465936\pi\)
−0.670834 + 0.741608i \(0.734064\pi\)
\(338\) 0 0
\(339\) 14.3090 10.3961i 0.777159 0.564639i
\(340\) 0 0
\(341\) 5.90983 3.52671i 0.320035 0.190982i
\(342\) 0 0
\(343\) 2.66312 1.93487i 0.143795 0.104473i
\(344\) 0 0
\(345\) −0.465558 + 1.43284i −0.0250648 + 0.0771415i
\(346\) 0 0
\(347\) 9.47214 0.508491 0.254245 0.967140i \(-0.418173\pi\)
0.254245 + 0.967140i \(0.418173\pi\)
\(348\) 0 0
\(349\) 7.63525 + 23.4989i 0.408706 + 1.25787i 0.917761 + 0.397133i \(0.129995\pi\)
−0.509055 + 0.860734i \(0.670005\pi\)
\(350\) 0 0
\(351\) −8.35410 + 6.06961i −0.445909 + 0.323972i
\(352\) 0 0
\(353\) −7.59017 + 5.51458i −0.403984 + 0.293512i −0.771162 0.636639i \(-0.780324\pi\)
0.367178 + 0.930151i \(0.380324\pi\)
\(354\) 0 0
\(355\) −1.54508 4.75528i −0.0820046 0.252384i
\(356\) 0 0
\(357\) 0.954915 + 0.693786i 0.0505395 + 0.0367191i
\(358\) 0 0
\(359\) −6.97214 5.06555i −0.367975 0.267350i 0.388396 0.921493i \(-0.373029\pi\)
−0.756371 + 0.654143i \(0.773029\pi\)
\(360\) 0 0
\(361\) 8.03444 24.7275i 0.422865 1.30145i
\(362\) 0 0
\(363\) 6.54508 + 20.1437i 0.343528 + 1.05727i
\(364\) 0 0
\(365\) −0.0172209 + 0.0530006i −0.000901385 + 0.00277418i
\(366\) 0 0
\(367\) −17.3820 −0.907331 −0.453666 0.891172i \(-0.649884\pi\)
−0.453666 + 0.891172i \(0.649884\pi\)
\(368\) 0 0
\(369\) −16.9443 12.3107i −0.882084 0.640871i
\(370\) 0 0
\(371\) −2.16718 −0.112515
\(372\) 0 0
\(373\) −18.7082 −0.968674 −0.484337 0.874881i \(-0.660939\pi\)
−0.484337 + 0.874881i \(0.660939\pi\)
\(374\) 0 0
\(375\) 6.80902 + 4.94704i 0.351616 + 0.255464i
\(376\) 0 0
\(377\) 25.9443 1.33620
\(378\) 0 0
\(379\) 3.92705 12.0862i 0.201719 0.620827i −0.798113 0.602508i \(-0.794168\pi\)
0.999832 0.0183198i \(-0.00583170\pi\)
\(380\) 0 0
\(381\) 6.21885 + 19.1396i 0.318601 + 0.980554i
\(382\) 0 0
\(383\) −2.89919 + 8.92278i −0.148142 + 0.455933i −0.997402 0.0720420i \(-0.977048\pi\)
0.849260 + 0.527975i \(0.177048\pi\)
\(384\) 0 0
\(385\) −0.0901699 0.0655123i −0.00459549 0.00333882i
\(386\) 0 0
\(387\) −1.85410 1.34708i −0.0942493 0.0684761i
\(388\) 0 0
\(389\) 8.39919 + 25.8500i 0.425856 + 1.31065i 0.902173 + 0.431375i \(0.141972\pi\)
−0.476317 + 0.879274i \(0.658028\pi\)
\(390\) 0 0
\(391\) 3.19098 2.31838i 0.161375 0.117246i
\(392\) 0 0
\(393\) 25.3885 18.4459i 1.28068 0.930470i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 28.1803 1.41433 0.707165 0.707048i \(-0.249974\pi\)
0.707165 + 0.707048i \(0.249974\pi\)
\(398\) 0 0
\(399\) 1.09424 3.36771i 0.0547803 0.168596i
\(400\) 0 0
\(401\) −17.7533 + 12.8985i −0.886557 + 0.644121i −0.934978 0.354706i \(-0.884581\pi\)
0.0484211 + 0.998827i \(0.484581\pi\)
\(402\) 0 0
\(403\) 5.70820 + 25.0705i 0.284346 + 1.24885i
\(404\) 0 0
\(405\) −3.39919 + 2.46965i −0.168907 + 0.122718i
\(406\) 0 0
\(407\) −1.79837 + 5.53483i −0.0891421 + 0.274351i
\(408\) 0 0
\(409\) 4.76393 0.235561 0.117781 0.993040i \(-0.462422\pi\)
0.117781 + 0.993040i \(0.462422\pi\)
\(410\) 0 0
\(411\) −7.23607 22.2703i −0.356929 1.09851i
\(412\) 0 0
\(413\) 2.42705 1.76336i 0.119427 0.0867691i
\(414\) 0 0
\(415\) 2.42705 1.76336i 0.119139 0.0865597i
\(416\) 0 0
\(417\) −9.73607 29.9645i −0.476777 1.46737i
\(418\) 0 0
\(419\) 4.85410 + 3.52671i 0.237138 + 0.172291i 0.700007 0.714136i \(-0.253180\pi\)
−0.462869 + 0.886427i \(0.653180\pi\)
\(420\) 0 0
\(421\) −1.38197 1.00406i −0.0673529 0.0489347i 0.553599 0.832783i \(-0.313254\pi\)
−0.620952 + 0.783848i \(0.713254\pi\)
\(422\) 0 0
\(423\) −5.29180 + 16.2865i −0.257296 + 0.791875i
\(424\) 0 0
\(425\) −3.35410 10.3229i −0.162698 0.500732i
\(426\) 0 0
\(427\) 0.145898 0.449028i 0.00706050 0.0217300i
\(428\) 0 0
\(429\) 12.7639 0.616248
\(430\) 0 0
\(431\) −0.145898 0.106001i −0.00702766 0.00510589i 0.584266 0.811562i \(-0.301383\pi\)
−0.591294 + 0.806456i \(0.701383\pi\)
\(432\) 0 0
\(433\) 19.4164 0.933093 0.466547 0.884497i \(-0.345498\pi\)
0.466547 + 0.884497i \(0.345498\pi\)
\(434\) 0 0
\(435\) 4.79837 0.230064
\(436\) 0 0
\(437\) −9.57295 6.95515i −0.457936 0.332710i
\(438\) 0 0
\(439\) −12.7082 −0.606529 −0.303265 0.952906i \(-0.598077\pi\)
−0.303265 + 0.952906i \(0.598077\pi\)
\(440\) 0 0
\(441\) −4.29180 + 13.2088i −0.204371 + 0.628990i
\(442\) 0 0
\(443\) −1.14590 3.52671i −0.0544433 0.167559i 0.920138 0.391595i \(-0.128077\pi\)
−0.974581 + 0.224036i \(0.928077\pi\)
\(444\) 0 0
\(445\) 1.89261 5.82485i 0.0897183 0.276124i
\(446\) 0 0
\(447\) 10.0623 + 7.31069i 0.475931 + 0.345784i
\(448\) 0 0
\(449\) −6.61803 4.80828i −0.312324 0.226917i 0.420569 0.907261i \(-0.361831\pi\)
−0.732893 + 0.680344i \(0.761831\pi\)
\(450\) 0 0
\(451\) −4.00000 12.3107i −0.188353 0.579690i
\(452\) 0 0
\(453\) 9.73607 7.07367i 0.457440 0.332350i
\(454\) 0 0
\(455\) 0.336881 0.244758i 0.0157932 0.0114745i
\(456\) 0 0
\(457\) −7.15248 22.0131i −0.334579 1.02973i −0.966929 0.255045i \(-0.917910\pi\)
0.632350 0.774682i \(-0.282090\pi\)
\(458\) 0 0
\(459\) 5.00000 0.233380
\(460\) 0 0
\(461\) 6.84346 21.0620i 0.318732 0.980955i −0.655459 0.755230i \(-0.727525\pi\)
0.974191 0.225725i \(-0.0724750\pi\)
\(462\) 0 0
\(463\) 2.61803 1.90211i 0.121670 0.0883987i −0.525286 0.850926i \(-0.676042\pi\)
0.646956 + 0.762527i \(0.276042\pi\)
\(464\) 0 0
\(465\) 1.05573 + 4.63677i 0.0489582 + 0.215025i
\(466\) 0 0
\(467\) −26.3713 + 19.1599i −1.22032 + 0.886614i −0.996127 0.0879310i \(-0.971975\pi\)
−0.224193 + 0.974545i \(0.571975\pi\)
\(468\) 0 0
\(469\) 0.218847 0.673542i 0.0101054 0.0311013i
\(470\) 0 0
\(471\) −44.0689 −2.03059
\(472\) 0 0
\(473\) −0.437694 1.34708i −0.0201252 0.0619390i
\(474\) 0 0
\(475\) −26.3435 + 19.1396i −1.20872 + 0.878187i
\(476\) 0 0
\(477\) 14.8541 10.7921i 0.680123 0.494138i
\(478\) 0 0
\(479\) 4.65248 + 14.3188i 0.212577 + 0.654245i 0.999317 + 0.0369600i \(0.0117674\pi\)
−0.786740 + 0.617285i \(0.788233\pi\)
\(480\) 0 0
\(481\) −17.5902 12.7800i −0.802043 0.582718i
\(482\) 0 0
\(483\) −0.753289 0.547296i −0.0342758 0.0249028i
\(484\) 0 0
\(485\) −0.555728 + 1.71036i −0.0252343 + 0.0776632i
\(486\) 0 0
\(487\) −4.71885 14.5231i −0.213831 0.658105i −0.999234 0.0391207i \(-0.987544\pi\)
0.785403 0.618985i \(-0.212456\pi\)
\(488\) 0 0
\(489\) 5.48936 16.8945i 0.248237 0.763996i
\(490\) 0 0
\(491\) −36.0689 −1.62777 −0.813883 0.581029i \(-0.802650\pi\)
−0.813883 + 0.581029i \(0.802650\pi\)
\(492\) 0 0
\(493\) −10.1631 7.38394i −0.457724 0.332556i
\(494\) 0 0
\(495\) 0.944272 0.0424419
\(496\) 0 0
\(497\) 3.09017 0.138613
\(498\) 0 0
\(499\) 15.0172 + 10.9106i 0.672263 + 0.488428i 0.870782 0.491669i \(-0.163613\pi\)
−0.198519 + 0.980097i \(0.563613\pi\)
\(500\) 0 0
\(501\) −41.7082 −1.86339
\(502\) 0 0
\(503\) −7.22542 + 22.2376i −0.322166 + 0.991524i 0.650538 + 0.759474i \(0.274543\pi\)
−0.972704 + 0.232051i \(0.925457\pi\)
\(504\) 0 0
\(505\) 1.09017 + 3.35520i 0.0485119 + 0.149304i
\(506\) 0 0
\(507\) −5.75329 + 17.7068i −0.255513 + 0.786387i
\(508\) 0 0
\(509\) −1.40983 1.02430i −0.0624896 0.0454014i 0.556102 0.831114i \(-0.312296\pi\)
−0.618592 + 0.785713i \(0.712296\pi\)
\(510\) 0 0
\(511\) −0.0278640 0.0202444i −0.00123263 0.000895560i
\(512\) 0 0
\(513\) −4.63525 14.2658i −0.204652 0.629853i
\(514\) 0 0
\(515\) 2.20820 1.60435i 0.0973051 0.0706963i
\(516\) 0 0
\(517\) −8.56231 + 6.22088i −0.376570 + 0.273594i
\(518\) 0 0
\(519\) −6.64590 20.4540i −0.291723 0.897830i
\(520\) 0 0
\(521\) −27.6525 −1.21148 −0.605738 0.795664i \(-0.707122\pi\)
−0.605738 + 0.795664i \(0.707122\pi\)
\(522\) 0 0
\(523\) 11.7254 36.0871i 0.512717 1.57798i −0.274681 0.961535i \(-0.588572\pi\)
0.787398 0.616445i \(-0.211428\pi\)
\(524\) 0 0
\(525\) −2.07295 + 1.50609i −0.0904709 + 0.0657310i
\(526\) 0 0
\(527\) 4.89919 11.4454i 0.213412 0.498571i
\(528\) 0 0
\(529\) 16.0902 11.6902i 0.699573 0.508269i
\(530\) 0 0
\(531\) −7.85410 + 24.1724i −0.340839 + 1.04899i
\(532\) 0 0
\(533\) 48.3607 2.09473
\(534\) 0 0
\(535\) 0.482779 + 1.48584i 0.0208724 + 0.0642385i
\(536\) 0 0
\(537\) 13.3541 9.70232i 0.576272 0.418686i
\(538\) 0 0
\(539\) −6.94427 + 5.04531i −0.299111 + 0.217317i
\(540\) 0 0
\(541\) −3.96556 12.2047i −0.170493 0.524722i 0.828906 0.559387i \(-0.188964\pi\)
−0.999399 + 0.0346650i \(0.988964\pi\)
\(542\) 0 0
\(543\) 29.6976 + 21.5765i 1.27444 + 0.925938i
\(544\) 0 0
\(545\) 2.16312 + 1.57160i 0.0926578 + 0.0673199i
\(546\) 0 0
\(547\) 1.58359 4.87380i 0.0677095 0.208388i −0.911477 0.411351i \(-0.865057\pi\)
0.979186 + 0.202963i \(0.0650570\pi\)
\(548\) 0 0
\(549\) 1.23607 + 3.80423i 0.0527541 + 0.162360i
\(550\) 0 0
\(551\) −11.6459 + 35.8424i −0.496132 + 1.52694i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −3.25329 2.36365i −0.138094 0.100331i
\(556\) 0 0
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 0 0
\(559\) 5.29180 0.223819
\(560\) 0 0
\(561\) −5.00000 3.63271i −0.211100 0.153373i
\(562\) 0 0
\(563\) −4.61803 −0.194627 −0.0973135 0.995254i \(-0.531025\pi\)
−0.0973135 + 0.995254i \(0.531025\pi\)
\(564\) 0 0
\(565\) −0.933629 + 2.87341i −0.0392781 + 0.120885i
\(566\) 0 0
\(567\) −0.802439 2.46965i −0.0336993 0.103716i
\(568\) 0 0
\(569\) −4.14590 + 12.7598i −0.173805 + 0.534917i −0.999577 0.0290873i \(-0.990740\pi\)
0.825772 + 0.564004i \(0.190740\pi\)
\(570\) 0 0
\(571\) −24.8435 18.0498i −1.03967 0.755362i −0.0694460 0.997586i \(-0.522123\pi\)
−0.970220 + 0.242224i \(0.922123\pi\)
\(572\) 0 0
\(573\) 38.6803 + 28.1029i 1.61589 + 1.17402i
\(574\) 0 0
\(575\) 2.64590 + 8.14324i 0.110342 + 0.339596i
\(576\) 0 0
\(577\) 17.3992 12.6412i 0.724338 0.526262i −0.163429 0.986555i \(-0.552256\pi\)
0.887767 + 0.460293i \(0.152256\pi\)
\(578\) 0 0
\(579\) 20.5902 14.9596i 0.855698 0.621701i
\(580\) 0 0
\(581\) 0.572949 + 1.76336i 0.0237699 + 0.0731563i
\(582\) 0 0
\(583\) 11.3475 0.469966
\(584\) 0 0
\(585\) −1.09017 + 3.35520i −0.0450730 + 0.138720i
\(586\) 0 0
\(587\) −0.0450850 + 0.0327561i −0.00186086 + 0.00135199i −0.588715 0.808340i \(-0.700366\pi\)
0.586854 + 0.809692i \(0.300366\pi\)
\(588\) 0 0
\(589\) −37.1976 3.36771i −1.53270 0.138764i
\(590\) 0 0
\(591\) 48.6418 35.3404i 2.00086 1.45371i
\(592\) 0 0
\(593\) −13.8885 + 42.7445i −0.570334 + 1.75531i 0.0812097 + 0.996697i \(0.474122\pi\)
−0.651544 + 0.758611i \(0.725878\pi\)
\(594\) 0 0
\(595\) −0.201626 −0.00826587
\(596\) 0 0
\(597\) 15.1631 + 46.6673i 0.620585 + 1.90996i
\(598\) 0 0
\(599\) 17.3541 12.6085i 0.709069 0.515169i −0.173804 0.984780i \(-0.555606\pi\)
0.882873 + 0.469611i \(0.155606\pi\)
\(600\) 0 0
\(601\) 24.2705 17.6336i 0.990015 0.719288i 0.0300904 0.999547i \(-0.490420\pi\)
0.959924 + 0.280259i \(0.0904205\pi\)
\(602\) 0 0
\(603\) 1.85410 + 5.70634i 0.0755049 + 0.232380i
\(604\) 0 0
\(605\) −2.92705 2.12663i −0.119002 0.0864597i
\(606\) 0 0
\(607\) 20.0902 + 14.5964i 0.815435 + 0.592448i 0.915401 0.402543i \(-0.131874\pi\)
−0.0999664 + 0.994991i \(0.531874\pi\)
\(608\) 0 0
\(609\) −0.916408 + 2.82041i −0.0371347 + 0.114289i
\(610\) 0 0
\(611\) −12.2188 37.6057i −0.494322 1.52137i
\(612\) 0 0
\(613\) −9.85410 + 30.3278i −0.398003 + 1.22493i 0.528594 + 0.848875i \(0.322719\pi\)
−0.926598 + 0.376054i \(0.877281\pi\)
\(614\) 0 0
\(615\) 8.94427 0.360668
\(616\) 0 0
\(617\) 1.71885 + 1.24882i 0.0691982 + 0.0502754i 0.621847 0.783139i \(-0.286383\pi\)
−0.552648 + 0.833415i \(0.686383\pi\)
\(618\) 0 0
\(619\) 38.8328 1.56082 0.780411 0.625266i \(-0.215010\pi\)
0.780411 + 0.625266i \(0.215010\pi\)
\(620\) 0 0
\(621\) −3.94427 −0.158278
\(622\) 0 0
\(623\) 3.06231 + 2.22490i 0.122689 + 0.0891386i
\(624\) 0 0
\(625\) 22.8328 0.913313
\(626\) 0 0
\(627\) −5.72949 + 17.6336i −0.228814 + 0.704216i
\(628\) 0 0
\(629\) 3.25329 + 10.0126i 0.129717 + 0.399228i
\(630\) 0 0
\(631\) −7.68034 + 23.6377i −0.305750 + 0.941000i 0.673647 + 0.739053i \(0.264727\pi\)
−0.979396 + 0.201947i \(0.935273\pi\)
\(632\) 0 0
\(633\) −17.8885 12.9968i −0.711006 0.516576i
\(634\) 0 0
\(635\) −2.78115 2.02063i −0.110367 0.0801861i
\(636\) 0 0
\(637\) −9.90983 30.4993i −0.392642 1.20843i
\(638\) 0 0
\(639\) −21.1803 + 15.3884i −0.837881 + 0.608756i
\(640\) 0 0
\(641\) −36.0517 + 26.1931i −1.42396 + 1.03456i −0.432852 + 0.901465i \(0.642493\pi\)
−0.991103 + 0.133099i \(0.957507\pi\)
\(642\) 0 0
\(643\) −8.38854 25.8173i −0.330812 1.01813i −0.968748 0.248045i \(-0.920212\pi\)
0.637937 0.770089i \(-0.279788\pi\)
\(644\) 0 0
\(645\) 0.978714 0.0385368
\(646\) 0 0
\(647\) 4.29180 13.2088i 0.168728 0.519291i −0.830564 0.556924i \(-0.811982\pi\)
0.999292 + 0.0376325i \(0.0119816\pi\)
\(648\) 0 0
\(649\) −12.7082 + 9.23305i −0.498841 + 0.362429i
\(650\) 0 0
\(651\) −2.92705 0.265003i −0.114720 0.0103863i
\(652\) 0 0
\(653\) 7.89919 5.73910i 0.309119 0.224588i −0.422399 0.906410i \(-0.638812\pi\)
0.731518 + 0.681822i \(0.238812\pi\)
\(654\) 0 0
\(655\) −1.65654 + 5.09831i −0.0647264 + 0.199207i
\(656\) 0 0
\(657\) 0.291796 0.0113840
\(658\) 0 0
\(659\) −2.91641 8.97578i −0.113607 0.349647i 0.878047 0.478575i \(-0.158846\pi\)
−0.991654 + 0.128928i \(0.958846\pi\)
\(660\) 0 0
\(661\) −31.9615 + 23.2214i −1.24316 + 0.903207i −0.997805 0.0662266i \(-0.978904\pi\)
−0.245353 + 0.969434i \(0.578904\pi\)
\(662\) 0 0
\(663\) 18.6803 13.5721i 0.725484 0.527095i
\(664\) 0 0
\(665\) 0.186918 + 0.575274i 0.00724836 + 0.0223082i
\(666\) 0 0
\(667\) 8.01722 + 5.82485i 0.310428 + 0.225539i
\(668\) 0 0
\(669\) −6.28115 4.56352i −0.242843 0.176436i
\(670\) 0 0
\(671\) −0.763932 + 2.35114i −0.0294913 + 0.0907648i
\(672\) 0 0
\(673\) −1.72542 5.31031i −0.0665102 0.204697i 0.912278 0.409571i \(-0.134322\pi\)
−0.978788 + 0.204874i \(0.934322\pi\)
\(674\) 0 0
\(675\) −3.35410 + 10.3229i −0.129099 + 0.397327i
\(676\) 0 0
\(677\) −30.0689 −1.15564 −0.577821 0.816164i \(-0.696097\pi\)
−0.577821 + 0.816164i \(0.696097\pi\)
\(678\) 0 0
\(679\) −0.899187 0.653298i −0.0345076 0.0250713i
\(680\) 0 0
\(681\) 54.3951 2.08443
\(682\) 0 0
\(683\) 36.5967 1.40034 0.700168 0.713978i \(-0.253109\pi\)
0.700168 + 0.713978i \(0.253109\pi\)
\(684\) 0 0
\(685\) 3.23607 + 2.35114i 0.123644 + 0.0898325i
\(686\) 0 0
\(687\) −46.1803 −1.76189
\(688\) 0 0
\(689\) −13.1008 + 40.3202i −0.499101 + 1.53608i
\(690\) 0 0
\(691\) 0.0385072 + 0.118513i 0.00146488 + 0.00450844i 0.951786 0.306762i \(-0.0992455\pi\)
−0.950321 + 0.311270i \(0.899246\pi\)
\(692\) 0 0
\(693\) −0.180340 + 0.555029i −0.00685055 + 0.0210838i
\(694\) 0 0
\(695\) 4.35410 + 3.16344i 0.165161 + 0.119996i
\(696\) 0 0
\(697\) −18.9443 13.7638i −0.717565 0.521342i
\(698\) 0 0
\(699\) −13.8820 42.7243i −0.525064 1.61598i
\(700\) 0 0
\(701\) −2.63525 + 1.91462i −0.0995322 + 0.0723144i −0.636438 0.771328i \(-0.719593\pi\)
0.536906 + 0.843642i \(0.319593\pi\)
\(702\) 0 0
\(703\) 25.5517 18.5644i 0.963699 0.700168i
\(704\) 0 0
\(705\) −2.25987 6.95515i −0.0851115 0.261946i
\(706\) 0 0
\(707\) −2.18034 −0.0820001
\(708\) 0 0
\(709\) 10.7705 33.1482i 0.404495 1.24491i −0.516821 0.856093i \(-0.672885\pi\)
0.921316 0.388814i \(-0.127115\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −3.86475 + 9.02878i −0.144736 + 0.338131i
\(714\) 0 0
\(715\) −1.76393 + 1.28157i −0.0659673 + 0.0479281i
\(716\) 0 0
\(717\) 2.03444 6.26137i 0.0759776 0.233835i
\(718\) 0 0
\(719\) 9.79837 0.365418 0.182709 0.983167i \(-0.441513\pi\)
0.182709 + 0.983167i \(0.441513\pi\)
\(720\) 0 0
\(721\) 0.521286 + 1.60435i 0.0194137 + 0.0597493i
\(722\) 0 0
\(723\) −17.1353 + 12.4495i −0.637267 + 0.463002i
\(724\) 0 0
\(725\) 22.0623 16.0292i 0.819373 0.595310i
\(726\) 0 0
\(727\) 3.12868 + 9.62908i 0.116036 + 0.357123i 0.992162 0.124960i \(-0.0398804\pi\)
−0.876126 + 0.482083i \(0.839880\pi\)
\(728\) 0 0
\(729\) 5.66312 + 4.11450i 0.209745 + 0.152389i
\(730\) 0 0
\(731\) −2.07295 1.50609i −0.0766708 0.0557046i
\(732\) 0 0
\(733\) −10.9615 + 33.7360i −0.404872 + 1.24607i 0.516130 + 0.856510i \(0.327372\pi\)
−0.921002 + 0.389558i \(0.872628\pi\)
\(734\) 0 0
\(735\) −1.83282 5.64083i −0.0676044 0.208065i
\(736\) 0 0
\(737\) −1.14590 + 3.52671i −0.0422097 + 0.129908i
\(738\) 0 0
\(739\) 4.29180 0.157876 0.0789381 0.996880i \(-0.474847\pi\)
0.0789381 + 0.996880i \(0.474847\pi\)
\(740\) 0 0
\(741\) −56.0410 40.7162i −2.05872 1.49575i
\(742\) 0 0
\(743\) 20.0344 0.734992 0.367496 0.930025i \(-0.380215\pi\)
0.367496 + 0.930025i \(0.380215\pi\)
\(744\) 0 0
\(745\) −2.12461 −0.0778398
\(746\) 0 0
\(747\) −12.7082 9.23305i −0.464969 0.337820i
\(748\) 0 0
\(749\) −0.965558 −0.0352807
\(750\) 0 0
\(751\) 13.9615 42.9691i 0.509462 1.56796i −0.283675 0.958920i \(-0.591554\pi\)
0.793137 0.609043i \(-0.208446\pi\)
\(752\) 0 0
\(753\) 0.652476 + 2.00811i 0.0237775 + 0.0731797i
\(754\) 0 0
\(755\) −0.635255 + 1.95511i −0.0231193 + 0.0711539i
\(756\) 0 0
\(757\) 27.2705 + 19.8132i 0.991164 + 0.720122i 0.960176 0.279397i \(-0.0901346\pi\)
0.0309880 + 0.999520i \(0.490135\pi\)
\(758\) 0 0
\(759\) 3.94427 + 2.86568i 0.143168 + 0.104018i
\(760\) 0 0
\(761\) 14.6418 + 45.0629i 0.530766 + 1.63353i 0.752624 + 0.658451i \(0.228788\pi\)
−0.221858 + 0.975079i \(0.571212\pi\)
\(762\) 0 0
\(763\) −1.33688 + 0.971301i −0.0483983 + 0.0351635i
\(764\) 0 0
\(765\) 1.38197 1.00406i 0.0499651 0.0363018i
\(766\) 0 0
\(767\) −18.1353 55.8146i −0.654826 2.01535i
\(768\) 0 0
\(769\) −20.2016 −0.728489 −0.364245 0.931303i \(-0.618673\pi\)
−0.364245 + 0.931303i \(0.618673\pi\)
\(770\) 0 0
\(771\) 17.5623 54.0512i 0.632491 1.94661i
\(772\) 0 0
\(773\) −0.545085 + 0.396027i −0.0196053 + 0.0142441i −0.597545 0.801836i \(-0.703857\pi\)
0.577939 + 0.816080i \(0.303857\pi\)
\(774\) 0 0
\(775\) 20.3435 + 17.7926i 0.730759 + 0.639128i
\(776\) 0 0
\(777\) 2.01064 1.46082i 0.0721314 0.0524065i
\(778\) 0 0
\(779\) −21.7082 + 66.8110i −0.777777 + 2.39375i
\(780\) 0 0
\(781\) −16.1803 −0.578978
\(782\) 0 0
\(783\) 3.88197 + 11.9475i 0.138730 + 0.426967i
\(784\) 0 0
\(785\) 6.09017 4.42477i 0.217367 0.157927i
\(786\) 0 0
\(787\) −20.0451 + 14.5636i −0.714530 + 0.519137i −0.884632 0.466290i \(-0.845590\pi\)
0.170102 + 0.985426i \(0.445590\pi\)
\(788\) 0 0
\(789\) −19.8607 61.1249i −0.707059 2.17610i
\(790\) 0 0
\(791\) −1.51064 1.09755i −0.0537123 0.0390243i
\(792\) 0 0
\(793\) −7.47214 5.42882i −0.265343 0.192783i
\(794\) 0 0
\(795\) −2.42299 + 7.45718i −0.0859344 + 0.264479i
\(796\) 0 0
\(797\) −13.8541 42.6385i −0.490737 1.51033i −0.823496 0.567322i \(-0.807979\pi\)
0.332758 0.943012i \(-0.392021\pi\)
\(798\) 0 0
\(799\) −5.91641 + 18.2088i −0.209307 + 0.644182i
\(800\) 0 0
\(801\) −32.0689 −1.13310
\(802\) 0 0
\(803\) 0.145898 + 0.106001i 0.00514863 + 0.00374070i
\(804\) 0 0
\(805\) 0.159054 0.00560590
\(806\) 0 0
\(807\) −27.6869 −0.974626
\(808\) 0 0
\(809\) −20.7254 15.0579i −0.728667 0.529408i 0.160474 0.987040i \(-0.448698\pi\)
−0.889142 + 0.457632i \(0.848698\pi\)
\(810\) 0 0
\(811\) −17.2918 −0.607197 −0.303599 0.952800i \(-0.598188\pi\)
−0.303599 + 0.952800i \(0.598188\pi\)
\(812\) 0 0
\(813\) −13.0279 + 40.0956i −0.456907 + 1.40622i
\(814\) 0 0
\(815\) 0.937694 + 2.88593i 0.0328460 + 0.101090i
\(816\) 0 0
\(817\) −2.37539 + 7.31069i −0.0831043 + 0.255769i
\(818\) 0 0
\(819\) −1.76393 1.28157i −0.0616368 0.0447817i
\(820\) 0 0
\(821\) 18.8541 + 13.6983i 0.658013 + 0.478074i 0.865991 0.500059i \(-0.166688\pi\)
−0.207979 + 0.978133i \(0.566688\pi\)
\(822\) 0 0
\(823\) −0.489357 1.50609i −0.0170579 0.0524988i 0.942165 0.335149i \(-0.108787\pi\)
−0.959223 + 0.282650i \(0.908787\pi\)
\(824\) 0 0
\(825\) 10.8541 7.88597i 0.377891 0.274554i
\(826\) 0 0
\(827\) 15.5000 11.2614i 0.538988 0.391598i −0.284721 0.958610i \(-0.591901\pi\)
0.823709 + 0.567013i \(0.191901\pi\)
\(828\) 0 0
\(829\) 2.56231 + 7.88597i 0.0889926 + 0.273891i 0.985642 0.168851i \(-0.0540057\pi\)
−0.896649 + 0.442742i \(0.854006\pi\)
\(830\) 0 0
\(831\) 29.9230 1.03802
\(832\) 0 0
\(833\) −4.79837 + 14.7679i −0.166254 + 0.511677i
\(834\) 0 0
\(835\) 5.76393 4.18774i 0.199469 0.144923i
\(836\) 0 0
\(837\) −10.6910 + 6.37988i −0.369534 + 0.220521i
\(838\) 0 0
\(839\) −23.3713 + 16.9803i −0.806868 + 0.586224i −0.912921 0.408137i \(-0.866179\pi\)
0.106053 + 0.994360i \(0.466179\pi\)
\(840\) 0 0
\(841\) 0.791796 2.43690i 0.0273033 0.0840310i
\(842\) 0 0
\(843\) 9.27051 0.319293
\(844\) 0 0
\(845\) −0.982779 3.02468i −0.0338086 0.104052i
\(846\) 0 0
\(847\) 1.80902 1.31433i 0.0621586 0.0451608i
\(848\) 0 0
\(849\) 34.3328 24.9443i 1.17830 0.856084i
\(850\) 0 0
\(851\) −2.56637 7.89848i −0.0879741 0.270756i
\(852\) 0 0
\(853\) −13.7082 9.95959i −0.469360 0.341010i 0.327832 0.944736i \(-0.393682\pi\)
−0.797192 + 0.603726i \(0.793682\pi\)
\(854\) 0 0
\(855\) −4.14590 3.01217i −0.141787 0.103014i
\(856\) 0 0
\(857\) 10.0902 31.0543i 0.344674 1.06080i −0.617085 0.786897i \(-0.711686\pi\)
0.961758 0.273900i \(-0.0883136\pi\)
\(858\) 0 0
\(859\) 7.07295 + 21.7683i 0.241326 + 0.742725i 0.996219 + 0.0868770i \(0.0276887\pi\)
−0.754893 + 0.655848i \(0.772311\pi\)
\(860\) 0 0
\(861\) −1.70820 + 5.25731i −0.0582154 + 0.179169i
\(862\) 0 0
\(863\) −40.6180 −1.38265 −0.691327 0.722542i \(-0.742974\pi\)
−0.691327 + 0.722542i \(0.742974\pi\)
\(864\) 0 0
\(865\) 2.97214 + 2.15938i 0.101056 + 0.0734212i
\(866\) 0 0
\(867\) 26.8328 0.911290
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −11.2082 8.14324i −0.379776 0.275923i
\(872\) 0 0
\(873\) 9.41641 0.318697
\(874\) 0 0
\(875\) 0.274575 0.845055i 0.00928233 0.0285681i
\(876\) 0 0
\(877\) 3.18034 + 9.78808i 0.107392 + 0.330520i 0.990285 0.139056i \(-0.0444068\pi\)
−0.882892 + 0.469576i \(0.844407\pi\)
\(878\) 0 0
\(879\) −18.7041 + 57.5654i −0.630875 + 1.94163i
\(880\) 0 0
\(881\) 13.1910 + 9.58381i 0.444416 + 0.322887i 0.787387 0.616459i \(-0.211433\pi\)
−0.342971 + 0.939346i \(0.611433\pi\)
\(882\) 0 0
\(883\) 37.4058 + 27.1769i 1.25880 + 0.914575i 0.998698 0.0510053i \(-0.0162425\pi\)
0.260106 + 0.965580i \(0.416243\pi\)
\(884\) 0 0
\(885\) −3.35410 10.3229i −0.112747 0.346999i
\(886\) 0 0
\(887\) 2.59017 1.88187i 0.0869694 0.0631870i −0.543451 0.839441i \(-0.682883\pi\)
0.630420 + 0.776254i \(0.282883\pi\)
\(888\) 0 0
\(889\) 1.71885 1.24882i 0.0576483 0.0418839i
\(890\) 0 0
\(891\) 4.20163 + 12.9313i 0.140760 + 0.433214i
\(892\) 0 0
\(893\) 57.4377 1.92208
\(894\) 0 0
\(895\) −0.871323 + 2.68166i −0.0291251 + 0.0896379i
\(896\) 0 0
\(897\) −14.7361 + 10.7064i −0.492023 + 0.357476i
\(898\) 0 0
\(899\) 31.1525 + 2.82041i 1.03899 + 0.0940661i
\(900\) 0 0
\(901\) 16.6074 12.0660i 0.553272 0.401976i
\(902\) 0 0
\(903\) −0.186918 + 0.575274i −0.00622023 + 0.0191439i
\(904\) 0 0
\(905\) −6.27051 −0.208439
\(906\) 0 0
\(907\) 15.7877 + 48.5896i 0.524223 + 1.61339i 0.765847 + 0.643023i \(0.222320\pi\)
−0.241624 + 0.970370i \(0.577680\pi\)
\(908\) 0 0
\(909\) 14.9443 10.8576i 0.495670 0.360125i
\(910\) 0 0
\(911\) 8.23607 5.98385i 0.272873 0.198254i −0.442930 0.896556i \(-0.646061\pi\)
0.715803 + 0.698302i \(0.246061\pi\)
\(912\) 0 0
\(913\) −3.00000 9.23305i −0.0992855 0.305569i
\(914\) 0 0
\(915\) −1.38197 1.00406i −0.0456864 0.0331931i
\(916\) 0 0
\(917\) −2.68034 1.94738i −0.0885126 0.0643082i
\(918\) 0 0
\(919\) 8.16312 25.1235i 0.269276 0.828748i −0.721401 0.692518i \(-0.756501\pi\)
0.990677 0.136230i \(-0.0434986\pi\)
\(920\) 0 0
\(921\) 17.8647 + 54.9820i 0.588663 + 1.81172i
\(922\) 0 0
\(923\) 18.6803 57.4922i 0.614871 1.89238i
\(924\) 0 0
\(925\) −22.8541 −0.751438
\(926\) 0 0
\(927\) −11.5623 8.40051i −0.379756 0.275909i
\(928\) 0 0
\(929\) 22.8197 0.748689 0.374344 0.927290i \(-0.377868\pi\)
0.374344 + 0.927290i \(0.377868\pi\)
\(930\) 0 0
\(931\) 46.5836 1.52672
\(932\) 0 0
\(933\) 34.2705 + 24.8990i 1.12197 + 0.815156i
\(934\) 0 0
\(935\) 1.05573 0.0345260
\(936\) 0 0
\(937\) −16.9828 + 52.2676i −0.554803 + 1.70751i 0.141659 + 0.989915i \(0.454756\pi\)
−0.696462 + 0.717593i \(0.745244\pi\)
\(938\) 0 0
\(939\) −9.87539 30.3933i −0.322271 0.991849i
\(940\) 0 0
\(941\) 17.1246 52.7041i 0.558246 1.71811i −0.128966 0.991649i \(-0.541166\pi\)
0.687212 0.726457i \(-0.258834\pi\)
\(942\) 0 0
\(943\) 14.9443 + 10.8576i 0.486652 + 0.353574i
\(944\) 0 0
\(945\) 0.163119 + 0.118513i 0.00530626 + 0.00385522i
\(946\) 0 0
\(947\) −0.742646 2.28563i −0.0241327 0.0742730i 0.938265 0.345918i \(-0.112432\pi\)
−0.962398 + 0.271645i \(0.912432\pi\)
\(948\) 0 0
\(949\) −0.545085 + 0.396027i −0.0176942 + 0.0128556i
\(950\) 0 0
\(951\) 8.94427 6.49839i 0.290038 0.210725i
\(952\) 0 0
\(953\) 1.73607 + 5.34307i 0.0562368 + 0.173079i 0.975229 0.221196i \(-0.0709959\pi\)
−0.918993 + 0.394275i \(0.870996\pi\)
\(954\) 0 0
\(955\) −8.16718 −0.264284
\(956\) 0 0
\(957\) 4.79837 14.7679i 0.155109 0.477378i
\(958\) 0 0
\(959\) −2.00000 + 1.45309i −0.0645834 + 0.0469226i
\(960\) 0 0
\(961\) 4.12868 + 30.7238i 0.133183 + 0.991091i
\(962\) 0 0
\(963\) 6.61803 4.80828i 0.213263 0.154945i
\(964\) 0 0
\(965\) −1.34346 + 4.13474i −0.0432475 + 0.133102i
\(966\) 0 0
\(967\) −13.1115 −0.421636 −0.210818 0.977525i \(-0.567613\pi\)
−0.210818 + 0.977525i \(0.567613\pi\)
\(968\) 0 0
\(969\) 10.3647 + 31.8994i 0.332964 + 1.02476i
\(970\) 0 0
\(971\) 2.02786 1.47333i 0.0650773 0.0472814i −0.554771 0.832003i \(-0.687194\pi\)
0.619848 + 0.784722i \(0.287194\pi\)
\(972\) 0 0
\(973\) −2.69098 + 1.95511i −0.0862690 + 0.0626781i
\(974\) 0 0
\(975\) 15.4894 + 47.6713i 0.496056 + 1.52670i
\(976\) 0 0
\(977\) 22.7082 + 16.4985i 0.726500 + 0.527833i 0.888454 0.458965i \(-0.151780\pi\)
−0.161954 + 0.986798i \(0.551780\pi\)
\(978\) 0 0
\(979\) −16.0344 11.6497i −0.512463 0.372326i
\(980\) 0 0
\(981\) 4.32624 13.3148i 0.138126 0.425109i
\(982\) 0 0
\(983\) −15.3713 47.3081i −0.490269 1.50889i −0.824202 0.566296i \(-0.808376\pi\)
0.333933 0.942597i \(-0.391624\pi\)
\(984\) 0 0
\(985\) −3.17376 + 9.76784i −0.101124 + 0.311229i
\(986\) 0 0
\(987\) 4.51973 0.143865
\(988\) 0 0
\(989\) 1.63525 + 1.18808i 0.0519981 + 0.0377788i
\(990\) 0 0
\(991\) −15.9787 −0.507581 −0.253790 0.967259i \(-0.581677\pi\)
−0.253790 + 0.967259i \(0.581677\pi\)
\(992\) 0 0
\(993\) −34.6738 −1.10034
\(994\) 0 0
\(995\) −6.78115 4.92680i −0.214977 0.156190i
\(996\) 0 0
\(997\) −42.8885 −1.35829 −0.679147 0.734002i \(-0.737650\pi\)
−0.679147 + 0.734002i \(0.737650\pi\)
\(998\) 0 0
\(999\) 3.25329 10.0126i 0.102930 0.316784i
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 124.2.f.b.109.1 yes 4
3.2 odd 2 1116.2.m.c.109.1 4
4.3 odd 2 496.2.n.a.481.1 4
31.2 even 5 inner 124.2.f.b.33.1 4
31.8 even 5 3844.2.a.g.1.1 2
31.23 odd 10 3844.2.a.f.1.2 2
93.2 odd 10 1116.2.m.c.901.1 4
124.95 odd 10 496.2.n.a.33.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
124.2.f.b.33.1 4 31.2 even 5 inner
124.2.f.b.109.1 yes 4 1.1 even 1 trivial
496.2.n.a.33.1 4 124.95 odd 10
496.2.n.a.481.1 4 4.3 odd 2
1116.2.m.c.109.1 4 3.2 odd 2
1116.2.m.c.901.1 4 93.2 odd 10
3844.2.a.f.1.2 2 31.23 odd 10
3844.2.a.g.1.1 2 31.8 even 5