Properties

Label 76.3.j.a.53.2
Level $76$
Weight $3$
Character 76.53
Analytic conductor $2.071$
Analytic rank $0$
Dimension $18$
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] = [76,3,Mod(13,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.13");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 76.j (of order \(18\), degree \(6\), 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: \(2.07085000914\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 93 x^{16} + 3429 x^{14} + 64261 x^{12} + 647217 x^{10} + 3386277 x^{8} + 8232133 x^{6} + 8319228 x^{4} + 2467872 x^{2} + 69312 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 53.2
Root \(-1.29756i\) of defining polynomial
Character \(\chi\) \(=\) 76.53
Dual form 76.3.j.a.33.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.95150 + 0.344102i) q^{3} +(-6.26982 + 5.26101i) q^{5} +(-0.733695 + 1.27080i) q^{7} +(-4.76730 + 1.73516i) q^{9} +O(q^{10})\) \(q+(-1.95150 + 0.344102i) q^{3} +(-6.26982 + 5.26101i) q^{5} +(-0.733695 + 1.27080i) q^{7} +(-4.76730 + 1.73516i) q^{9} +(4.00699 + 6.94030i) q^{11} +(3.08826 + 0.544543i) q^{13} +(10.4252 - 12.4243i) q^{15} +(-24.4418 - 8.89608i) q^{17} +(18.9886 - 0.657543i) q^{19} +(0.994520 - 2.73242i) q^{21} +(15.0102 + 12.5950i) q^{23} +(7.29129 - 41.3510i) q^{25} +(24.1514 - 13.9438i) q^{27} +(14.4735 + 39.7655i) q^{29} +(-23.5291 - 13.5845i) q^{31} +(-10.2078 - 12.1652i) q^{33} +(-2.08553 - 11.8276i) q^{35} +34.6395i q^{37} -6.21410 q^{39} +(-41.3997 + 7.29988i) q^{41} +(-25.7244 + 21.5853i) q^{43} +(20.7615 - 35.9599i) q^{45} +(13.2418 - 4.81962i) q^{47} +(23.4234 + 40.5705i) q^{49} +(50.7592 + 8.95022i) q^{51} +(-11.9055 + 14.1884i) q^{53} +(-61.6361 - 22.4337i) q^{55} +(-36.8300 + 7.81721i) q^{57} +(-38.5848 + 106.011i) q^{59} +(5.25904 + 4.41286i) q^{61} +(1.29271 - 7.33134i) q^{63} +(-22.2277 + 12.8331i) q^{65} +(-32.1119 - 88.2267i) q^{67} +(-33.6263 - 19.4142i) q^{69} +(40.4679 + 48.2277i) q^{71} +(-11.1289 - 63.1153i) q^{73} +83.2052i q^{75} -11.7596 q^{77} +(131.967 - 23.2694i) q^{79} +(-7.35616 + 6.17255i) q^{81} +(64.9155 - 112.437i) q^{83} +(200.048 - 72.8115i) q^{85} +(-41.9283 - 72.6219i) q^{87} +(27.7532 + 4.89363i) q^{89} +(-2.95784 + 3.52502i) q^{91} +(50.5915 + 18.4138i) q^{93} +(-115.596 + 104.022i) q^{95} +(28.3963 - 78.0182i) q^{97} +(-31.1450 - 26.1338i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 6 q^{3} + 9 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 6 q^{3} + 9 q^{7} + 6 q^{9} - 15 q^{11} + 51 q^{13} + 21 q^{15} - 45 q^{17} + 30 q^{19} - 63 q^{21} + 48 q^{23} - 54 q^{25} - 198 q^{27} - 39 q^{29} - 108 q^{31} - 105 q^{33} + 51 q^{35} + 48 q^{39} + 54 q^{41} + 75 q^{43} + 288 q^{45} + 339 q^{47} - 24 q^{49} + 360 q^{51} + 69 q^{53} - 51 q^{55} + 510 q^{57} - 483 q^{59} - 36 q^{61} - 267 q^{63} - 585 q^{65} - 87 q^{67} - 351 q^{69} - 234 q^{71} - 132 q^{73} + 108 q^{77} + 363 q^{79} + 258 q^{81} + 279 q^{83} + 666 q^{85} + 600 q^{89} + 270 q^{91} - 456 q^{93} - 39 q^{95} - 801 q^{97} - 267 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(e\left(\frac{11}{18}\right)\) \(1\)

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
\(3\) −1.95150 + 0.344102i −0.650499 + 0.114701i −0.489154 0.872198i \(-0.662694\pi\)
−0.161345 + 0.986898i \(0.551583\pi\)
\(4\) 0 0
\(5\) −6.26982 + 5.26101i −1.25396 + 1.05220i −0.257667 + 0.966234i \(0.582954\pi\)
−0.996298 + 0.0859675i \(0.972602\pi\)
\(6\) 0 0
\(7\) −0.733695 + 1.27080i −0.104814 + 0.181542i −0.913662 0.406475i \(-0.866758\pi\)
0.808848 + 0.588017i \(0.200091\pi\)
\(8\) 0 0
\(9\) −4.76730 + 1.73516i −0.529700 + 0.192795i
\(10\) 0 0
\(11\) 4.00699 + 6.94030i 0.364271 + 0.630937i 0.988659 0.150178i \(-0.0479847\pi\)
−0.624388 + 0.781115i \(0.714651\pi\)
\(12\) 0 0
\(13\) 3.08826 + 0.544543i 0.237558 + 0.0418879i 0.291160 0.956674i \(-0.405959\pi\)
−0.0536014 + 0.998562i \(0.517070\pi\)
\(14\) 0 0
\(15\) 10.4252 12.4243i 0.695015 0.828286i
\(16\) 0 0
\(17\) −24.4418 8.89608i −1.43775 0.523299i −0.498610 0.866827i \(-0.666156\pi\)
−0.939143 + 0.343528i \(0.888378\pi\)
\(18\) 0 0
\(19\) 18.9886 0.657543i 0.999401 0.0346075i
\(20\) 0 0
\(21\) 0.994520 2.73242i 0.0473581 0.130115i
\(22\) 0 0
\(23\) 15.0102 + 12.5950i 0.652617 + 0.547611i 0.907864 0.419265i \(-0.137712\pi\)
−0.255247 + 0.966876i \(0.582157\pi\)
\(24\) 0 0
\(25\) 7.29129 41.3510i 0.291652 1.65404i
\(26\) 0 0
\(27\) 24.1514 13.9438i 0.894495 0.516437i
\(28\) 0 0
\(29\) 14.4735 + 39.7655i 0.499085 + 1.37122i 0.892161 + 0.451718i \(0.149189\pi\)
−0.393076 + 0.919506i \(0.628589\pi\)
\(30\) 0 0
\(31\) −23.5291 13.5845i −0.759004 0.438211i 0.0699340 0.997552i \(-0.477721\pi\)
−0.828938 + 0.559340i \(0.811054\pi\)
\(32\) 0 0
\(33\) −10.2078 12.1652i −0.309327 0.368641i
\(34\) 0 0
\(35\) −2.08553 11.8276i −0.0595867 0.337933i
\(36\) 0 0
\(37\) 34.6395i 0.936202i 0.883675 + 0.468101i \(0.155062\pi\)
−0.883675 + 0.468101i \(0.844938\pi\)
\(38\) 0 0
\(39\) −6.21410 −0.159336
\(40\) 0 0
\(41\) −41.3997 + 7.29988i −1.00975 + 0.178046i −0.653967 0.756524i \(-0.726896\pi\)
−0.355781 + 0.934569i \(0.615785\pi\)
\(42\) 0 0
\(43\) −25.7244 + 21.5853i −0.598242 + 0.501985i −0.890880 0.454239i \(-0.849911\pi\)
0.292638 + 0.956223i \(0.405467\pi\)
\(44\) 0 0
\(45\) 20.7615 35.9599i 0.461366 0.799109i
\(46\) 0 0
\(47\) 13.2418 4.81962i 0.281740 0.102545i −0.197285 0.980346i \(-0.563212\pi\)
0.479025 + 0.877801i \(0.340990\pi\)
\(48\) 0 0
\(49\) 23.4234 + 40.5705i 0.478028 + 0.827969i
\(50\) 0 0
\(51\) 50.7592 + 8.95022i 0.995279 + 0.175495i
\(52\) 0 0
\(53\) −11.9055 + 14.1884i −0.224631 + 0.267705i −0.866575 0.499046i \(-0.833684\pi\)
0.641944 + 0.766752i \(0.278128\pi\)
\(54\) 0 0
\(55\) −61.6361 22.4337i −1.12066 0.407885i
\(56\) 0 0
\(57\) −36.8300 + 7.81721i −0.646140 + 0.137144i
\(58\) 0 0
\(59\) −38.5848 + 106.011i −0.653980 + 1.79680i −0.0514857 + 0.998674i \(0.516396\pi\)
−0.602495 + 0.798123i \(0.705827\pi\)
\(60\) 0 0
\(61\) 5.25904 + 4.41286i 0.0862137 + 0.0723419i 0.684876 0.728660i \(-0.259856\pi\)
−0.598662 + 0.801002i \(0.704301\pi\)
\(62\) 0 0
\(63\) 1.29271 7.33134i 0.0205193 0.116371i
\(64\) 0 0
\(65\) −22.2277 + 12.8331i −0.341964 + 0.197433i
\(66\) 0 0
\(67\) −32.1119 88.2267i −0.479282 1.31682i −0.910104 0.414380i \(-0.863999\pi\)
0.430822 0.902437i \(-0.358224\pi\)
\(68\) 0 0
\(69\) −33.6263 19.4142i −0.487338 0.281365i
\(70\) 0 0
\(71\) 40.4679 + 48.2277i 0.569970 + 0.679264i 0.971625 0.236527i \(-0.0760092\pi\)
−0.401655 + 0.915791i \(0.631565\pi\)
\(72\) 0 0
\(73\) −11.1289 63.1153i −0.152451 0.864593i −0.961079 0.276273i \(-0.910901\pi\)
0.808628 0.588320i \(-0.200211\pi\)
\(74\) 0 0
\(75\) 83.2052i 1.10940i
\(76\) 0 0
\(77\) −11.7596 −0.152722
\(78\) 0 0
\(79\) 131.967 23.2694i 1.67047 0.294549i 0.743235 0.669031i \(-0.233291\pi\)
0.927234 + 0.374482i \(0.122179\pi\)
\(80\) 0 0
\(81\) −7.35616 + 6.17255i −0.0908168 + 0.0762043i
\(82\) 0 0
\(83\) 64.9155 112.437i 0.782114 1.35466i −0.148593 0.988898i \(-0.547475\pi\)
0.930708 0.365764i \(-0.119192\pi\)
\(84\) 0 0
\(85\) 200.048 72.8115i 2.35351 0.856606i
\(86\) 0 0
\(87\) −41.9283 72.6219i −0.481934 0.834735i
\(88\) 0 0
\(89\) 27.7532 + 4.89363i 0.311833 + 0.0549846i 0.327375 0.944895i \(-0.393836\pi\)
−0.0155415 + 0.999879i \(0.504947\pi\)
\(90\) 0 0
\(91\) −2.95784 + 3.52502i −0.0325037 + 0.0387365i
\(92\) 0 0
\(93\) 50.5915 + 18.4138i 0.543994 + 0.197998i
\(94\) 0 0
\(95\) −115.596 + 104.022i −1.21680 + 1.09497i
\(96\) 0 0
\(97\) 28.3963 78.0182i 0.292746 0.804312i −0.702917 0.711272i \(-0.748119\pi\)
0.995662 0.0930397i \(-0.0296584\pi\)
\(98\) 0 0
\(99\) −31.1450 26.1338i −0.314596 0.263977i
\(100\) 0 0
\(101\) 14.3170 81.1957i 0.141752 0.803918i −0.828165 0.560484i \(-0.810615\pi\)
0.969917 0.243434i \(-0.0782739\pi\)
\(102\) 0 0
\(103\) −86.8134 + 50.1218i −0.842849 + 0.486619i −0.858232 0.513263i \(-0.828437\pi\)
0.0153828 + 0.999882i \(0.495103\pi\)
\(104\) 0 0
\(105\) 8.13982 + 22.3640i 0.0775221 + 0.212990i
\(106\) 0 0
\(107\) −108.773 62.7998i −1.01657 0.586914i −0.103458 0.994634i \(-0.532991\pi\)
−0.913107 + 0.407719i \(0.866324\pi\)
\(108\) 0 0
\(109\) 103.327 + 123.141i 0.947958 + 1.12973i 0.991425 + 0.130677i \(0.0417152\pi\)
−0.0434671 + 0.999055i \(0.513840\pi\)
\(110\) 0 0
\(111\) −11.9195 67.5989i −0.107383 0.608999i
\(112\) 0 0
\(113\) 172.214i 1.52402i 0.647565 + 0.762010i \(0.275787\pi\)
−0.647565 + 0.762010i \(0.724213\pi\)
\(114\) 0 0
\(115\) −160.374 −1.39456
\(116\) 0 0
\(117\) −15.6675 + 2.76261i −0.133910 + 0.0236120i
\(118\) 0 0
\(119\) 29.2379 24.5335i 0.245697 0.206164i
\(120\) 0 0
\(121\) 28.3881 49.1697i 0.234613 0.406361i
\(122\) 0 0
\(123\) 78.2794 28.4914i 0.636418 0.231637i
\(124\) 0 0
\(125\) 69.5242 + 120.419i 0.556194 + 0.963356i
\(126\) 0 0
\(127\) 66.6088 + 11.7449i 0.524479 + 0.0924798i 0.429618 0.903011i \(-0.358648\pi\)
0.0948610 + 0.995491i \(0.469759\pi\)
\(128\) 0 0
\(129\) 42.7735 50.9755i 0.331578 0.395159i
\(130\) 0 0
\(131\) −68.0952 24.7846i −0.519811 0.189196i 0.0687722 0.997632i \(-0.478092\pi\)
−0.588583 + 0.808437i \(0.700314\pi\)
\(132\) 0 0
\(133\) −13.0962 + 24.6131i −0.0984680 + 0.185061i
\(134\) 0 0
\(135\) −78.0664 + 214.486i −0.578269 + 1.58878i
\(136\) 0 0
\(137\) −123.653 103.757i −0.902576 0.757351i 0.0681164 0.997677i \(-0.478301\pi\)
−0.970692 + 0.240326i \(0.922746\pi\)
\(138\) 0 0
\(139\) −33.1861 + 188.208i −0.238749 + 1.35401i 0.595825 + 0.803115i \(0.296825\pi\)
−0.834573 + 0.550897i \(0.814286\pi\)
\(140\) 0 0
\(141\) −24.1829 + 13.9620i −0.171510 + 0.0990212i
\(142\) 0 0
\(143\) 8.59531 + 23.6154i 0.0601070 + 0.165143i
\(144\) 0 0
\(145\) −299.953 173.178i −2.06864 1.19433i
\(146\) 0 0
\(147\) −59.6710 71.1132i −0.405925 0.483763i
\(148\) 0 0
\(149\) 9.13688 + 51.8178i 0.0613214 + 0.347771i 0.999996 + 0.00296543i \(0.000943928\pi\)
−0.938674 + 0.344805i \(0.887945\pi\)
\(150\) 0 0
\(151\) 15.0198i 0.0994687i −0.998762 0.0497344i \(-0.984163\pi\)
0.998762 0.0497344i \(-0.0158375\pi\)
\(152\) 0 0
\(153\) 131.957 0.862467
\(154\) 0 0
\(155\) 218.992 38.6142i 1.41285 0.249124i
\(156\) 0 0
\(157\) −173.728 + 145.775i −1.10654 + 0.928501i −0.997848 0.0655736i \(-0.979112\pi\)
−0.108697 + 0.994075i \(0.534668\pi\)
\(158\) 0 0
\(159\) 18.3512 31.7853i 0.115417 0.199907i
\(160\) 0 0
\(161\) −27.0186 + 9.83398i −0.167818 + 0.0610806i
\(162\) 0 0
\(163\) 101.688 + 176.129i 0.623853 + 1.08054i 0.988761 + 0.149502i \(0.0477669\pi\)
−0.364909 + 0.931043i \(0.618900\pi\)
\(164\) 0 0
\(165\) 128.002 + 22.5702i 0.775770 + 0.136789i
\(166\) 0 0
\(167\) 131.582 156.814i 0.787918 0.939004i −0.211344 0.977412i \(-0.567784\pi\)
0.999262 + 0.0384078i \(0.0122286\pi\)
\(168\) 0 0
\(169\) −149.567 54.4380i −0.885013 0.322119i
\(170\) 0 0
\(171\) −89.3835 + 36.0829i −0.522711 + 0.211011i
\(172\) 0 0
\(173\) 25.3925 69.7654i 0.146778 0.403268i −0.844416 0.535688i \(-0.820052\pi\)
0.991194 + 0.132420i \(0.0422746\pi\)
\(174\) 0 0
\(175\) 47.1991 + 39.6047i 0.269709 + 0.226313i
\(176\) 0 0
\(177\) 38.8197 220.157i 0.219320 1.24383i
\(178\) 0 0
\(179\) 197.109 113.801i 1.10117 0.635761i 0.164642 0.986353i \(-0.447353\pi\)
0.936528 + 0.350593i \(0.114020\pi\)
\(180\) 0 0
\(181\) 113.203 + 311.022i 0.625429 + 1.71835i 0.693292 + 0.720657i \(0.256160\pi\)
−0.0678628 + 0.997695i \(0.521618\pi\)
\(182\) 0 0
\(183\) −11.7815 6.80203i −0.0643796 0.0371696i
\(184\) 0 0
\(185\) −182.239 217.184i −0.985073 1.17396i
\(186\) 0 0
\(187\) −36.1964 205.280i −0.193564 1.09775i
\(188\) 0 0
\(189\) 40.9220i 0.216518i
\(190\) 0 0
\(191\) 173.982 0.910900 0.455450 0.890261i \(-0.349478\pi\)
0.455450 + 0.890261i \(0.349478\pi\)
\(192\) 0 0
\(193\) −153.848 + 27.1275i −0.797139 + 0.140557i −0.557361 0.830270i \(-0.688186\pi\)
−0.239779 + 0.970828i \(0.577075\pi\)
\(194\) 0 0
\(195\) 38.9613 32.6924i 0.199802 0.167653i
\(196\) 0 0
\(197\) 9.29665 16.1023i 0.0471911 0.0817374i −0.841465 0.540312i \(-0.818306\pi\)
0.888656 + 0.458574i \(0.151640\pi\)
\(198\) 0 0
\(199\) 32.2054 11.7218i 0.161836 0.0589035i −0.259832 0.965654i \(-0.583667\pi\)
0.421668 + 0.906750i \(0.361445\pi\)
\(200\) 0 0
\(201\) 93.0252 + 161.124i 0.462812 + 0.801614i
\(202\) 0 0
\(203\) −61.1530 10.7829i −0.301246 0.0531178i
\(204\) 0 0
\(205\) 221.164 263.573i 1.07885 1.28572i
\(206\) 0 0
\(207\) −93.4124 33.9993i −0.451268 0.164248i
\(208\) 0 0
\(209\) 80.6507 + 129.152i 0.385888 + 0.617952i
\(210\) 0 0
\(211\) 53.0635 145.791i 0.251486 0.690952i −0.748138 0.663543i \(-0.769052\pi\)
0.999624 0.0274090i \(-0.00872566\pi\)
\(212\) 0 0
\(213\) −95.5682 80.1912i −0.448677 0.376485i
\(214\) 0 0
\(215\) 47.7269 270.672i 0.221985 1.25894i
\(216\) 0 0
\(217\) 34.5264 19.9338i 0.159108 0.0918609i
\(218\) 0 0
\(219\) 43.4362 + 119.340i 0.198339 + 0.544931i
\(220\) 0 0
\(221\) −70.6382 40.7830i −0.319630 0.184538i
\(222\) 0 0
\(223\) 239.061 + 284.901i 1.07202 + 1.27758i 0.958820 + 0.284013i \(0.0916658\pi\)
0.113201 + 0.993572i \(0.463890\pi\)
\(224\) 0 0
\(225\) 36.9906 + 209.784i 0.164403 + 0.932374i
\(226\) 0 0
\(227\) 64.3587i 0.283518i 0.989901 + 0.141759i \(0.0452759\pi\)
−0.989901 + 0.141759i \(0.954724\pi\)
\(228\) 0 0
\(229\) −110.363 −0.481936 −0.240968 0.970533i \(-0.577465\pi\)
−0.240968 + 0.970533i \(0.577465\pi\)
\(230\) 0 0
\(231\) 22.9489 4.04650i 0.0993457 0.0175173i
\(232\) 0 0
\(233\) 74.0516 62.1367i 0.317818 0.266681i −0.469896 0.882722i \(-0.655709\pi\)
0.787714 + 0.616041i \(0.211264\pi\)
\(234\) 0 0
\(235\) −57.6677 + 99.8833i −0.245394 + 0.425035i
\(236\) 0 0
\(237\) −249.526 + 90.8201i −1.05285 + 0.383207i
\(238\) 0 0
\(239\) −97.2196 168.389i −0.406777 0.704558i 0.587750 0.809043i \(-0.300014\pi\)
−0.994526 + 0.104485i \(0.966681\pi\)
\(240\) 0 0
\(241\) 44.1394 + 7.78296i 0.183151 + 0.0322945i 0.264471 0.964394i \(-0.414803\pi\)
−0.0813203 + 0.996688i \(0.525914\pi\)
\(242\) 0 0
\(243\) −149.101 + 177.691i −0.613583 + 0.731239i
\(244\) 0 0
\(245\) −360.302 131.139i −1.47062 0.535262i
\(246\) 0 0
\(247\) 58.9998 + 8.30946i 0.238866 + 0.0336415i
\(248\) 0 0
\(249\) −87.9927 + 241.758i −0.353384 + 0.970915i
\(250\) 0 0
\(251\) 65.2741 + 54.7715i 0.260056 + 0.218213i 0.763488 0.645822i \(-0.223485\pi\)
−0.503432 + 0.864035i \(0.667930\pi\)
\(252\) 0 0
\(253\) −27.2678 + 154.643i −0.107778 + 0.611239i
\(254\) 0 0
\(255\) −365.339 + 210.928i −1.43270 + 0.827170i
\(256\) 0 0
\(257\) 56.5131 + 155.268i 0.219895 + 0.604157i 0.999763 0.0217875i \(-0.00693573\pi\)
−0.779867 + 0.625945i \(0.784714\pi\)
\(258\) 0 0
\(259\) −44.0197 25.4148i −0.169960 0.0981267i
\(260\) 0 0
\(261\) −137.999 164.460i −0.528731 0.630116i
\(262\) 0 0
\(263\) −0.962817 5.46041i −0.00366090 0.0207620i 0.982923 0.184020i \(-0.0589110\pi\)
−0.986583 + 0.163258i \(0.947800\pi\)
\(264\) 0 0
\(265\) 151.593i 0.572051i
\(266\) 0 0
\(267\) −55.8441 −0.209154
\(268\) 0 0
\(269\) −27.2932 + 4.81253i −0.101462 + 0.0178905i −0.224149 0.974555i \(-0.571960\pi\)
0.122687 + 0.992445i \(0.460849\pi\)
\(270\) 0 0
\(271\) −307.622 + 258.126i −1.13514 + 0.952493i −0.999269 0.0382341i \(-0.987827\pi\)
−0.135868 + 0.990727i \(0.543382\pi\)
\(272\) 0 0
\(273\) 4.55925 7.89686i 0.0167006 0.0289262i
\(274\) 0 0
\(275\) 316.204 115.089i 1.14983 0.418505i
\(276\) 0 0
\(277\) −62.5547 108.348i −0.225829 0.391148i 0.730739 0.682657i \(-0.239176\pi\)
−0.956568 + 0.291510i \(0.905842\pi\)
\(278\) 0 0
\(279\) 135.742 + 23.9349i 0.486529 + 0.0857883i
\(280\) 0 0
\(281\) −121.544 + 144.851i −0.432542 + 0.515483i −0.937654 0.347570i \(-0.887007\pi\)
0.505112 + 0.863054i \(0.331451\pi\)
\(282\) 0 0
\(283\) 507.009 + 184.536i 1.79155 + 0.652071i 0.999113 + 0.0421181i \(0.0134106\pi\)
0.792438 + 0.609953i \(0.208812\pi\)
\(284\) 0 0
\(285\) 189.791 242.775i 0.665933 0.851843i
\(286\) 0 0
\(287\) 21.0981 57.9664i 0.0735124 0.201974i
\(288\) 0 0
\(289\) 296.874 + 249.107i 1.02725 + 0.861961i
\(290\) 0 0
\(291\) −28.5691 + 162.024i −0.0981757 + 0.556782i
\(292\) 0 0
\(293\) 197.211 113.860i 0.673076 0.388601i −0.124165 0.992262i \(-0.539625\pi\)
0.797241 + 0.603661i \(0.206292\pi\)
\(294\) 0 0
\(295\) −315.804 867.665i −1.07052 2.94124i
\(296\) 0 0
\(297\) 193.548 + 111.745i 0.651678 + 0.376246i
\(298\) 0 0
\(299\) 39.4968 + 47.0704i 0.132096 + 0.157426i
\(300\) 0 0
\(301\) −8.55671 48.5275i −0.0284276 0.161221i
\(302\) 0 0
\(303\) 163.380i 0.539207i
\(304\) 0 0
\(305\) −56.1893 −0.184227
\(306\) 0 0
\(307\) −315.954 + 55.7112i −1.02917 + 0.181470i −0.662640 0.748938i \(-0.730564\pi\)
−0.366526 + 0.930408i \(0.619453\pi\)
\(308\) 0 0
\(309\) 152.169 127.685i 0.492457 0.413220i
\(310\) 0 0
\(311\) 86.9979 150.685i 0.279736 0.484517i −0.691583 0.722297i \(-0.743086\pi\)
0.971319 + 0.237780i \(0.0764197\pi\)
\(312\) 0 0
\(313\) 10.4686 3.81027i 0.0334461 0.0121734i −0.325243 0.945630i \(-0.605446\pi\)
0.358689 + 0.933457i \(0.383224\pi\)
\(314\) 0 0
\(315\) 30.4652 + 52.7672i 0.0967148 + 0.167515i
\(316\) 0 0
\(317\) −166.576 29.3719i −0.525477 0.0926558i −0.0953845 0.995441i \(-0.530408\pi\)
−0.430093 + 0.902785i \(0.641519\pi\)
\(318\) 0 0
\(319\) −217.990 + 259.790i −0.683353 + 0.814389i
\(320\) 0 0
\(321\) 233.879 + 85.1249i 0.728594 + 0.265187i
\(322\) 0 0
\(323\) −469.965 152.853i −1.45500 0.473229i
\(324\) 0 0
\(325\) 45.0348 123.732i 0.138568 0.380714i
\(326\) 0 0
\(327\) −244.016 204.754i −0.746226 0.626158i
\(328\) 0 0
\(329\) −3.59068 + 20.3638i −0.0109139 + 0.0618959i
\(330\) 0 0
\(331\) 79.1289 45.6851i 0.239060 0.138022i −0.375684 0.926748i \(-0.622592\pi\)
0.614745 + 0.788726i \(0.289259\pi\)
\(332\) 0 0
\(333\) −60.1049 165.137i −0.180495 0.495906i
\(334\) 0 0
\(335\) 665.497 + 384.225i 1.98656 + 1.14694i
\(336\) 0 0
\(337\) 272.227 + 324.428i 0.807796 + 0.962694i 0.999825 0.0186950i \(-0.00595114\pi\)
−0.192029 + 0.981389i \(0.561507\pi\)
\(338\) 0 0
\(339\) −59.2592 336.076i −0.174806 0.991373i
\(340\) 0 0
\(341\) 217.732i 0.638511i
\(342\) 0 0
\(343\) −140.645 −0.410042
\(344\) 0 0
\(345\) 312.969 55.1849i 0.907157 0.159956i
\(346\) 0 0
\(347\) 442.043 370.918i 1.27390 1.06893i 0.279843 0.960046i \(-0.409718\pi\)
0.994055 0.108882i \(-0.0347269\pi\)
\(348\) 0 0
\(349\) −3.21731 + 5.57255i −0.00921866 + 0.0159672i −0.870598 0.491995i \(-0.836268\pi\)
0.861379 + 0.507962i \(0.169601\pi\)
\(350\) 0 0
\(351\) 82.1786 29.9106i 0.234127 0.0852153i
\(352\) 0 0
\(353\) −239.173 414.260i −0.677544 1.17354i −0.975718 0.219029i \(-0.929711\pi\)
0.298175 0.954511i \(-0.403622\pi\)
\(354\) 0 0
\(355\) −507.453 89.4777i −1.42945 0.252050i
\(356\) 0 0
\(357\) −48.6157 + 57.9379i −0.136178 + 0.162291i
\(358\) 0 0
\(359\) 341.910 + 124.445i 0.952395 + 0.346643i 0.771049 0.636776i \(-0.219732\pi\)
0.181346 + 0.983419i \(0.441955\pi\)
\(360\) 0 0
\(361\) 360.135 24.9717i 0.997605 0.0691736i
\(362\) 0 0
\(363\) −38.4800 + 105.723i −0.106005 + 0.291248i
\(364\) 0 0
\(365\) 401.827 + 337.173i 1.10089 + 0.923760i
\(366\) 0 0
\(367\) −42.2602 + 239.670i −0.115150 + 0.653051i 0.871525 + 0.490351i \(0.163131\pi\)
−0.986676 + 0.162700i \(0.947980\pi\)
\(368\) 0 0
\(369\) 184.698 106.636i 0.500537 0.288985i
\(370\) 0 0
\(371\) −9.29557 25.5394i −0.0250554 0.0688393i
\(372\) 0 0
\(373\) 127.166 + 73.4195i 0.340928 + 0.196835i 0.660682 0.750666i \(-0.270267\pi\)
−0.319754 + 0.947501i \(0.603600\pi\)
\(374\) 0 0
\(375\) −177.113 211.075i −0.472301 0.562866i
\(376\) 0 0
\(377\) 23.0437 + 130.687i 0.0611240 + 0.346651i
\(378\) 0 0
\(379\) 13.3180i 0.0351398i 0.999846 + 0.0175699i \(0.00559296\pi\)
−0.999846 + 0.0175699i \(0.994407\pi\)
\(380\) 0 0
\(381\) −134.028 −0.351780
\(382\) 0 0
\(383\) −60.2733 + 10.6278i −0.157372 + 0.0277488i −0.251779 0.967785i \(-0.581016\pi\)
0.0944072 + 0.995534i \(0.469904\pi\)
\(384\) 0 0
\(385\) 73.7307 61.8674i 0.191508 0.160695i
\(386\) 0 0
\(387\) 85.1820 147.540i 0.220109 0.381239i
\(388\) 0 0
\(389\) −630.808 + 229.595i −1.62162 + 0.590220i −0.983689 0.179878i \(-0.942430\pi\)
−0.637926 + 0.770098i \(0.720208\pi\)
\(390\) 0 0
\(391\) −254.829 441.377i −0.651737 1.12884i
\(392\) 0 0
\(393\) 141.416 + 24.9355i 0.359837 + 0.0634490i
\(394\) 0 0
\(395\) −704.990 + 840.174i −1.78478 + 2.12702i
\(396\) 0 0
\(397\) −330.666 120.352i −0.832911 0.303155i −0.109858 0.993947i \(-0.535040\pi\)
−0.723053 + 0.690793i \(0.757262\pi\)
\(398\) 0 0
\(399\) 17.0879 52.5388i 0.0428268 0.131676i
\(400\) 0 0
\(401\) −17.8166 + 48.9508i −0.0444305 + 0.122072i −0.959924 0.280262i \(-0.909579\pi\)
0.915493 + 0.402334i \(0.131801\pi\)
\(402\) 0 0
\(403\) −65.2666 54.7652i −0.161952 0.135894i
\(404\) 0 0
\(405\) 13.6480 77.4016i 0.0336988 0.191115i
\(406\) 0 0
\(407\) −240.409 + 138.800i −0.590684 + 0.341032i
\(408\) 0 0
\(409\) −23.5263 64.6379i −0.0575214 0.158039i 0.907604 0.419828i \(-0.137910\pi\)
−0.965125 + 0.261789i \(0.915687\pi\)
\(410\) 0 0
\(411\) 277.011 + 159.932i 0.673993 + 0.389130i
\(412\) 0 0
\(413\) −106.409 126.813i −0.257649 0.307054i
\(414\) 0 0
\(415\) 184.523 + 1046.48i 0.444633 + 2.52164i
\(416\) 0 0
\(417\) 378.706i 0.908168i
\(418\) 0 0
\(419\) −59.7382 −0.142573 −0.0712866 0.997456i \(-0.522711\pi\)
−0.0712866 + 0.997456i \(0.522711\pi\)
\(420\) 0 0
\(421\) −201.011 + 35.4437i −0.477462 + 0.0841894i −0.407200 0.913339i \(-0.633495\pi\)
−0.0702620 + 0.997529i \(0.522384\pi\)
\(422\) 0 0
\(423\) −54.7648 + 45.9531i −0.129468 + 0.108636i
\(424\) 0 0
\(425\) −546.074 + 945.828i −1.28488 + 2.22548i
\(426\) 0 0
\(427\) −9.46637 + 3.44548i −0.0221695 + 0.00806903i
\(428\) 0 0
\(429\) −24.8998 43.1277i −0.0580415 0.100531i
\(430\) 0 0
\(431\) −342.996 60.4795i −0.795815 0.140324i −0.239065 0.971004i \(-0.576841\pi\)
−0.556750 + 0.830680i \(0.687952\pi\)
\(432\) 0 0
\(433\) 351.516 418.921i 0.811816 0.967484i −0.188077 0.982154i \(-0.560225\pi\)
0.999892 + 0.0146700i \(0.00466976\pi\)
\(434\) 0 0
\(435\) 644.947 + 234.742i 1.48264 + 0.539636i
\(436\) 0 0
\(437\) 293.305 + 229.293i 0.671177 + 0.524697i
\(438\) 0 0
\(439\) 184.868 507.922i 0.421113 1.15700i −0.529958 0.848024i \(-0.677792\pi\)
0.951071 0.308974i \(-0.0999855\pi\)
\(440\) 0 0
\(441\) −182.062 152.769i −0.412840 0.346414i
\(442\) 0 0
\(443\) −20.9050 + 118.558i −0.0471896 + 0.267626i −0.999269 0.0382248i \(-0.987830\pi\)
0.952080 + 0.305851i \(0.0989408\pi\)
\(444\) 0 0
\(445\) −199.753 + 115.327i −0.448883 + 0.259163i
\(446\) 0 0
\(447\) −35.6612 97.9783i −0.0797789 0.219191i
\(448\) 0 0
\(449\) 179.289 + 103.512i 0.399307 + 0.230540i 0.686185 0.727427i \(-0.259284\pi\)
−0.286878 + 0.957967i \(0.592617\pi\)
\(450\) 0 0
\(451\) −216.551 258.076i −0.480158 0.572230i
\(452\) 0 0
\(453\) 5.16833 + 29.3111i 0.0114091 + 0.0647043i
\(454\) 0 0
\(455\) 37.6625i 0.0827746i
\(456\) 0 0
\(457\) 822.006 1.79870 0.899350 0.437229i \(-0.144040\pi\)
0.899350 + 0.437229i \(0.144040\pi\)
\(458\) 0 0
\(459\) −714.348 + 125.959i −1.55631 + 0.274420i
\(460\) 0 0
\(461\) −94.9044 + 79.6342i −0.205866 + 0.172742i −0.739891 0.672726i \(-0.765123\pi\)
0.534025 + 0.845469i \(0.320679\pi\)
\(462\) 0 0
\(463\) 20.7428 35.9276i 0.0448009 0.0775975i −0.842755 0.538297i \(-0.819068\pi\)
0.887556 + 0.460699i \(0.152401\pi\)
\(464\) 0 0
\(465\) −414.075 + 150.711i −0.890483 + 0.324109i
\(466\) 0 0
\(467\) 104.451 + 180.914i 0.223663 + 0.387396i 0.955917 0.293636i \(-0.0948652\pi\)
−0.732255 + 0.681031i \(0.761532\pi\)
\(468\) 0 0
\(469\) 135.679 + 23.9238i 0.289293 + 0.0510102i
\(470\) 0 0
\(471\) 288.867 344.259i 0.613307 0.730910i
\(472\) 0 0
\(473\) −252.886 92.0430i −0.534643 0.194594i
\(474\) 0 0
\(475\) 111.262 789.992i 0.234235 1.66314i
\(476\) 0 0
\(477\) 32.1379 88.2981i 0.0673750 0.185111i
\(478\) 0 0
\(479\) −169.213 141.986i −0.353263 0.296423i 0.448836 0.893614i \(-0.351839\pi\)
−0.802099 + 0.597192i \(0.796283\pi\)
\(480\) 0 0
\(481\) −18.8627 + 106.976i −0.0392156 + 0.222403i
\(482\) 0 0
\(483\) 49.3429 28.4881i 0.102159 0.0589816i
\(484\) 0 0
\(485\) 232.415 + 638.554i 0.479205 + 1.31661i
\(486\) 0 0
\(487\) 464.405 + 268.124i 0.953604 + 0.550563i 0.894199 0.447671i \(-0.147746\pi\)
0.0594052 + 0.998234i \(0.481080\pi\)
\(488\) 0 0
\(489\) −259.050 308.724i −0.529755 0.631337i
\(490\) 0 0
\(491\) −36.3136 205.945i −0.0739585 0.419440i −0.999197 0.0400548i \(-0.987247\pi\)
0.925239 0.379385i \(-0.123864\pi\)
\(492\) 0 0
\(493\) 1100.70i 2.23265i
\(494\) 0 0
\(495\) 332.764 0.672250
\(496\) 0 0
\(497\) −90.9787 + 16.0420i −0.183056 + 0.0322777i
\(498\) 0 0
\(499\) 190.517 159.863i 0.381798 0.320367i −0.431610 0.902061i \(-0.642054\pi\)
0.813408 + 0.581694i \(0.197610\pi\)
\(500\) 0 0
\(501\) −202.823 + 351.299i −0.404835 + 0.701196i
\(502\) 0 0
\(503\) −566.747 + 206.279i −1.12673 + 0.410098i −0.837105 0.547041i \(-0.815754\pi\)
−0.289629 + 0.957139i \(0.593532\pi\)
\(504\) 0 0
\(505\) 337.406 + 584.405i 0.668131 + 1.15724i
\(506\) 0 0
\(507\) 310.612 + 54.7693i 0.612647 + 0.108026i
\(508\) 0 0
\(509\) 408.323 486.620i 0.802205 0.956031i −0.197500 0.980303i \(-0.563282\pi\)
0.999705 + 0.0242718i \(0.00772670\pi\)
\(510\) 0 0
\(511\) 88.3720 + 32.1648i 0.172939 + 0.0629448i
\(512\) 0 0
\(513\) 449.432 280.654i 0.876086 0.547084i
\(514\) 0 0
\(515\) 280.614 770.981i 0.544881 1.49705i
\(516\) 0 0
\(517\) 86.5093 + 72.5899i 0.167329 + 0.140406i
\(518\) 0 0
\(519\) −25.5471 + 144.885i −0.0492236 + 0.279161i
\(520\) 0 0
\(521\) 742.280 428.556i 1.42472 0.822564i 0.428025 0.903767i \(-0.359210\pi\)
0.996698 + 0.0812032i \(0.0258763\pi\)
\(522\) 0 0
\(523\) 61.0403 + 167.707i 0.116712 + 0.320663i 0.984270 0.176673i \(-0.0565336\pi\)
−0.867558 + 0.497337i \(0.834311\pi\)
\(524\) 0 0
\(525\) −105.737 61.0472i −0.201404 0.116280i
\(526\) 0 0
\(527\) 454.245 + 541.348i 0.861944 + 1.02723i
\(528\) 0 0
\(529\) −25.1893 142.855i −0.0476167 0.270048i
\(530\) 0 0
\(531\) 572.337i 1.07785i
\(532\) 0 0
\(533\) −131.828 −0.247332
\(534\) 0 0
\(535\) 1012.37 178.509i 1.89229 0.333662i
\(536\) 0 0
\(537\) −345.499 + 289.908i −0.643388 + 0.539866i
\(538\) 0 0
\(539\) −187.714 + 325.131i −0.348264 + 0.603211i
\(540\) 0 0
\(541\) −783.678 + 285.236i −1.44857 + 0.527238i −0.942192 0.335072i \(-0.891239\pi\)
−0.506381 + 0.862310i \(0.669017\pi\)
\(542\) 0 0
\(543\) −327.938 568.005i −0.603937 1.04605i
\(544\) 0 0
\(545\) −1295.69 228.465i −2.37741 0.419202i
\(546\) 0 0
\(547\) −93.6553 + 111.614i −0.171216 + 0.204048i −0.844828 0.535038i \(-0.820297\pi\)
0.673612 + 0.739085i \(0.264742\pi\)
\(548\) 0 0
\(549\) −32.7284 11.9122i −0.0596146 0.0216979i
\(550\) 0 0
\(551\) 300.979 + 745.575i 0.546241 + 1.35313i
\(552\) 0 0
\(553\) −67.2529 + 184.776i −0.121615 + 0.334134i
\(554\) 0 0
\(555\) 430.371 + 361.124i 0.775444 + 0.650674i
\(556\) 0 0
\(557\) −40.1890 + 227.923i −0.0721526 + 0.409198i 0.927244 + 0.374458i \(0.122171\pi\)
−0.999396 + 0.0347397i \(0.988940\pi\)
\(558\) 0 0
\(559\) −91.1977 + 52.6530i −0.163144 + 0.0941914i
\(560\) 0 0
\(561\) 141.274 + 388.148i 0.251826 + 0.691886i
\(562\) 0 0
\(563\) −214.859 124.049i −0.381632 0.220335i 0.296896 0.954910i \(-0.404049\pi\)
−0.678528 + 0.734574i \(0.737382\pi\)
\(564\) 0 0
\(565\) −906.021 1079.75i −1.60358 1.91107i
\(566\) 0 0
\(567\) −2.44688 13.8770i −0.00431549 0.0244743i
\(568\) 0 0
\(569\) 113.410i 0.199314i −0.995022 0.0996570i \(-0.968225\pi\)
0.995022 0.0996570i \(-0.0317746\pi\)
\(570\) 0 0
\(571\) −70.0063 −0.122603 −0.0613015 0.998119i \(-0.519525\pi\)
−0.0613015 + 0.998119i \(0.519525\pi\)
\(572\) 0 0
\(573\) −339.525 + 59.8675i −0.592540 + 0.104481i
\(574\) 0 0
\(575\) 630.261 528.852i 1.09611 0.919742i
\(576\) 0 0
\(577\) −378.755 + 656.024i −0.656422 + 1.13696i 0.325114 + 0.945675i \(0.394597\pi\)
−0.981535 + 0.191281i \(0.938736\pi\)
\(578\) 0 0
\(579\) 290.899 105.879i 0.502416 0.182865i
\(580\) 0 0
\(581\) 95.2563 + 164.989i 0.163952 + 0.283974i
\(582\) 0 0
\(583\) −146.177 25.7749i −0.250732 0.0442108i
\(584\) 0 0
\(585\) 83.6985 99.7479i 0.143074 0.170509i
\(586\) 0 0
\(587\) 491.918 + 179.044i 0.838021 + 0.305015i 0.725147 0.688595i \(-0.241772\pi\)
0.112875 + 0.993609i \(0.463994\pi\)
\(588\) 0 0
\(589\) −455.718 242.480i −0.773715 0.411681i
\(590\) 0 0
\(591\) −12.6016 + 34.6225i −0.0213224 + 0.0585829i
\(592\) 0 0
\(593\) 78.0279 + 65.4732i 0.131582 + 0.110410i 0.706203 0.708009i \(-0.250406\pi\)
−0.574622 + 0.818419i \(0.694851\pi\)
\(594\) 0 0
\(595\) −54.2456 + 307.642i −0.0911690 + 0.517045i
\(596\) 0 0
\(597\) −58.8152 + 33.9570i −0.0985179 + 0.0568793i
\(598\) 0 0
\(599\) 40.2642 + 110.625i 0.0672191 + 0.184683i 0.968754 0.248023i \(-0.0797810\pi\)
−0.901535 + 0.432706i \(0.857559\pi\)
\(600\) 0 0
\(601\) −7.12203 4.11190i −0.0118503 0.00684177i 0.494063 0.869426i \(-0.335511\pi\)
−0.505913 + 0.862584i \(0.668844\pi\)
\(602\) 0 0
\(603\) 306.174 + 364.884i 0.507751 + 0.605115i
\(604\) 0 0
\(605\) 80.6935 + 457.635i 0.133378 + 0.756422i
\(606\) 0 0
\(607\) 13.7752i 0.0226940i −0.999936 0.0113470i \(-0.996388\pi\)
0.999936 0.0113470i \(-0.00361194\pi\)
\(608\) 0 0
\(609\) 123.050 0.202053
\(610\) 0 0
\(611\) 43.5185 7.67349i 0.0712251 0.0125589i
\(612\) 0 0
\(613\) 57.7119 48.4260i 0.0941466 0.0789984i −0.594500 0.804096i \(-0.702650\pi\)
0.688646 + 0.725098i \(0.258205\pi\)
\(614\) 0 0
\(615\) −340.905 + 590.464i −0.554317 + 0.960104i
\(616\) 0 0
\(617\) −178.372 + 64.9220i −0.289095 + 0.105222i −0.482497 0.875898i \(-0.660270\pi\)
0.193402 + 0.981120i \(0.438048\pi\)
\(618\) 0 0
\(619\) 70.1021 + 121.420i 0.113250 + 0.196156i 0.917079 0.398705i \(-0.130540\pi\)
−0.803829 + 0.594861i \(0.797207\pi\)
\(620\) 0 0
\(621\) 538.139 + 94.8884i 0.866569 + 0.152799i
\(622\) 0 0
\(623\) −26.5812 + 31.6782i −0.0426664 + 0.0508478i
\(624\) 0 0
\(625\) −83.0162 30.2154i −0.132826 0.0483447i
\(626\) 0 0
\(627\) −201.831 224.288i −0.321899 0.357716i
\(628\) 0 0
\(629\) 308.156 846.651i 0.489914 1.34603i
\(630\) 0 0
\(631\) −601.130 504.408i −0.952662 0.799378i 0.0270817 0.999633i \(-0.491379\pi\)
−0.979744 + 0.200255i \(0.935823\pi\)
\(632\) 0 0
\(633\) −53.3864 + 302.769i −0.0843388 + 0.478309i
\(634\) 0 0
\(635\) −479.416 + 276.791i −0.754986 + 0.435891i
\(636\) 0 0
\(637\) 50.2450 + 138.047i 0.0788776 + 0.216714i
\(638\) 0 0
\(639\) −276.605 159.698i −0.432872 0.249919i
\(640\) 0 0
\(641\) 182.035 + 216.940i 0.283985 + 0.338441i 0.889113 0.457688i \(-0.151322\pi\)
−0.605128 + 0.796129i \(0.706878\pi\)
\(642\) 0 0
\(643\) 131.759 + 747.245i 0.204913 + 1.16212i 0.897576 + 0.440860i \(0.145327\pi\)
−0.692662 + 0.721262i \(0.743562\pi\)
\(644\) 0 0
\(645\) 544.639i 0.844402i
\(646\) 0 0
\(647\) −980.479 −1.51542 −0.757712 0.652589i \(-0.773683\pi\)
−0.757712 + 0.652589i \(0.773683\pi\)
\(648\) 0 0
\(649\) −890.357 + 156.994i −1.37189 + 0.241901i
\(650\) 0 0
\(651\) −60.5189 + 50.7814i −0.0929629 + 0.0780052i
\(652\) 0 0
\(653\) 462.959 801.868i 0.708972 1.22798i −0.256266 0.966606i \(-0.582492\pi\)
0.965239 0.261370i \(-0.0841742\pi\)
\(654\) 0 0
\(655\) 557.337 202.854i 0.850897 0.309701i
\(656\) 0 0
\(657\) 162.570 + 281.579i 0.247443 + 0.428583i
\(658\) 0 0
\(659\) −721.793 127.272i −1.09528 0.193128i −0.403320 0.915059i \(-0.632144\pi\)
−0.691965 + 0.721931i \(0.743255\pi\)
\(660\) 0 0
\(661\) −690.154 + 822.494i −1.04411 + 1.24432i −0.0751278 + 0.997174i \(0.523936\pi\)
−0.968979 + 0.247144i \(0.920508\pi\)
\(662\) 0 0
\(663\) 151.884 + 55.2812i 0.229086 + 0.0833803i
\(664\) 0 0
\(665\) −47.3786 223.219i −0.0712460 0.335668i
\(666\) 0 0
\(667\) −283.599 + 779.182i −0.425186 + 1.16819i
\(668\) 0 0
\(669\) −564.561 473.723i −0.843888 0.708106i
\(670\) 0 0
\(671\) −9.55367 + 54.1816i −0.0142380 + 0.0807475i
\(672\) 0 0
\(673\) 8.17686 4.72091i 0.0121499 0.00701473i −0.493913 0.869511i \(-0.664434\pi\)
0.506063 + 0.862497i \(0.331100\pi\)
\(674\) 0 0
\(675\) −400.495 1100.35i −0.593326 1.63015i
\(676\) 0 0
\(677\) 959.523 + 553.981i 1.41732 + 0.818288i 0.996062 0.0886554i \(-0.0282570\pi\)
0.421253 + 0.906943i \(0.361590\pi\)
\(678\) 0 0
\(679\) 78.3111 + 93.3275i 0.115333 + 0.137448i
\(680\) 0 0
\(681\) −22.1459 125.596i −0.0325197 0.184428i
\(682\) 0 0
\(683\) 224.067i 0.328062i 0.986455 + 0.164031i \(0.0524498\pi\)
−0.986455 + 0.164031i \(0.947550\pi\)
\(684\) 0 0
\(685\) 1321.15 1.92868
\(686\) 0 0
\(687\) 215.374 37.9762i 0.313499 0.0552783i
\(688\) 0 0
\(689\) −44.4933 + 37.3343i −0.0645767 + 0.0541863i
\(690\) 0 0
\(691\) 613.870 1063.25i 0.888379 1.53872i 0.0465880 0.998914i \(-0.485165\pi\)
0.841791 0.539804i \(-0.181501\pi\)
\(692\) 0 0
\(693\) 56.0616 20.4048i 0.0808970 0.0294441i
\(694\) 0 0
\(695\) −782.091 1354.62i −1.12531 1.94909i
\(696\) 0 0
\(697\) 1076.82 + 189.873i 1.54494 + 0.272414i
\(698\) 0 0
\(699\) −123.130 + 146.741i −0.176152 + 0.209930i
\(700\) 0 0
\(701\) 589.300 + 214.488i 0.840657 + 0.305974i 0.726225 0.687457i \(-0.241273\pi\)
0.114432 + 0.993431i \(0.463495\pi\)
\(702\) 0 0
\(703\) 22.7770 + 657.756i 0.0323997 + 0.935642i
\(704\) 0 0
\(705\) 78.1683 214.766i 0.110877 0.304632i
\(706\) 0 0
\(707\) 92.6790 + 77.7669i 0.131088 + 0.109996i
\(708\) 0 0
\(709\) 38.8989 220.606i 0.0548644 0.311151i −0.945009 0.327043i \(-0.893948\pi\)
0.999874 + 0.0158920i \(0.00505878\pi\)
\(710\) 0 0
\(711\) −588.751 + 339.915i −0.828060 + 0.478081i
\(712\) 0 0
\(713\) −182.079 500.257i −0.255370 0.701623i
\(714\) 0 0
\(715\) −178.132 102.844i −0.249135 0.143838i
\(716\) 0 0
\(717\) 247.667 + 295.158i 0.345421 + 0.411657i
\(718\) 0 0
\(719\) −149.642 848.661i −0.208125 1.18034i −0.892446 0.451154i \(-0.851012\pi\)
0.684321 0.729181i \(-0.260099\pi\)
\(720\) 0 0
\(721\) 147.096i 0.204017i
\(722\) 0 0
\(723\) −88.8160 −0.122844
\(724\) 0 0
\(725\) 1749.87 308.550i 2.41362 0.425586i
\(726\) 0 0
\(727\) 678.448 569.286i 0.933216 0.783062i −0.0431756 0.999068i \(-0.513747\pi\)
0.976392 + 0.216006i \(0.0693030\pi\)
\(728\) 0 0
\(729\) 273.038 472.916i 0.374538 0.648719i
\(730\) 0 0
\(731\) 820.775 298.738i 1.12281 0.408670i
\(732\) 0 0
\(733\) 85.7061 + 148.447i 0.116925 + 0.202520i 0.918548 0.395310i \(-0.129363\pi\)
−0.801623 + 0.597831i \(0.796030\pi\)
\(734\) 0 0
\(735\) 748.254 + 131.937i 1.01803 + 0.179507i
\(736\) 0 0
\(737\) 483.648 576.389i 0.656239 0.782075i
\(738\) 0 0
\(739\) 246.083 + 89.5668i 0.332994 + 0.121200i 0.503106 0.864225i \(-0.332191\pi\)
−0.170112 + 0.985425i \(0.554413\pi\)
\(740\) 0 0
\(741\) −117.997 + 4.08604i −0.159240 + 0.00551422i
\(742\) 0 0
\(743\) −398.919 + 1096.02i −0.536904 + 1.47513i 0.313802 + 0.949488i \(0.398397\pi\)
−0.850706 + 0.525642i \(0.823825\pi\)
\(744\) 0 0
\(745\) −329.901 276.819i −0.442820 0.371570i
\(746\) 0 0
\(747\) −114.376 + 648.659i −0.153114 + 0.868352i
\(748\) 0 0
\(749\) 159.612 92.1518i 0.213100 0.123033i
\(750\) 0 0
\(751\) −350.713 963.577i −0.466995 1.28306i −0.920129 0.391616i \(-0.871916\pi\)
0.453134 0.891443i \(-0.350306\pi\)
\(752\) 0 0
\(753\) −146.229 84.4255i −0.194195 0.112119i
\(754\) 0 0
\(755\) 79.0192 + 94.1714i 0.104661 + 0.124730i
\(756\) 0 0
\(757\) −52.0633 295.266i −0.0687759 0.390047i −0.999692 0.0248145i \(-0.992100\pi\)
0.930916 0.365233i \(-0.119011\pi\)
\(758\) 0 0
\(759\) 311.169i 0.409972i
\(760\) 0 0
\(761\) −872.406 −1.14639 −0.573197 0.819417i \(-0.694297\pi\)
−0.573197 + 0.819417i \(0.694297\pi\)
\(762\) 0 0
\(763\) −232.298 + 40.9604i −0.304453 + 0.0536833i
\(764\) 0 0
\(765\) −827.350 + 694.229i −1.08150 + 0.907489i
\(766\) 0 0
\(767\) −176.887 + 306.378i −0.230622 + 0.399450i
\(768\) 0 0
\(769\) 192.314 69.9967i 0.250084 0.0910230i −0.213937 0.976848i \(-0.568629\pi\)
0.464020 + 0.885825i \(0.346406\pi\)
\(770\) 0 0
\(771\) −163.713 283.560i −0.212339 0.367782i
\(772\) 0 0
\(773\) −952.826 168.009i −1.23263 0.217347i −0.480878 0.876788i \(-0.659682\pi\)
−0.751757 + 0.659441i \(0.770793\pi\)
\(774\) 0 0
\(775\) −733.292 + 873.903i −0.946183 + 1.12762i
\(776\) 0 0
\(777\) 94.6497 + 34.4497i 0.121814 + 0.0443368i
\(778\) 0 0
\(779\) −781.322 + 165.837i −1.00298 + 0.212884i
\(780\) 0 0
\(781\) −172.561 + 474.107i −0.220949 + 0.607051i
\(782\) 0 0
\(783\) 904.036 + 758.576i 1.15458 + 0.968807i
\(784\) 0 0
\(785\) 322.319 1827.96i 0.410598 2.32862i
\(786\) 0 0
\(787\) 668.219 385.796i 0.849071 0.490211i −0.0112663 0.999937i \(-0.503586\pi\)
0.860337 + 0.509725i \(0.170253\pi\)
\(788\) 0 0
\(789\) 3.75787 + 10.3247i 0.00476282 + 0.0130857i
\(790\) 0 0
\(791\) −218.849 126.353i −0.276674 0.159738i
\(792\) 0 0
\(793\) 13.8383 + 16.4918i 0.0174505 + 0.0207967i
\(794\) 0 0
\(795\) 52.1635 + 295.834i 0.0656145 + 0.372118i
\(796\) 0 0
\(797\) 1437.98i 1.80424i 0.431488 + 0.902119i \(0.357989\pi\)
−0.431488 + 0.902119i \(0.642011\pi\)
\(798\) 0 0
\(799\) −366.529 −0.458735
\(800\) 0 0
\(801\) −140.799 + 24.8266i −0.175779 + 0.0309945i
\(802\) 0 0
\(803\) 393.446 330.140i 0.489970 0.411134i
\(804\) 0 0
\(805\) 117.665 203.803i 0.146168 0.253171i
\(806\) 0 0
\(807\) 51.6067 18.7833i 0.0639488 0.0232754i
\(808\) 0 0
\(809\) 230.478 + 399.199i 0.284892 + 0.493448i 0.972583 0.232556i \(-0.0747089\pi\)
−0.687691 + 0.726004i \(0.741376\pi\)
\(810\) 0 0
\(811\) 548.267 + 96.6743i 0.676038 + 0.119204i 0.501118 0.865379i \(-0.332922\pi\)
0.174920 + 0.984583i \(0.444033\pi\)
\(812\) 0 0
\(813\) 511.502 609.584i 0.629154 0.749796i
\(814\) 0 0
\(815\) −1564.18 569.315i −1.91924 0.698546i
\(816\) 0 0
\(817\) −474.278 + 426.791i −0.580511 + 0.522388i
\(818\) 0 0
\(819\) 7.98446 21.9371i 0.00974904 0.0267853i
\(820\) 0 0
\(821\) 1049.23 + 880.405i 1.27799 + 1.07236i 0.993518 + 0.113672i \(0.0362612\pi\)
0.284467 + 0.958686i \(0.408183\pi\)
\(822\) 0 0
\(823\) −58.8718 + 333.878i −0.0715331 + 0.405685i 0.927925 + 0.372767i \(0.121591\pi\)
−0.999458 + 0.0329174i \(0.989520\pi\)
\(824\) 0 0
\(825\) −577.469 + 333.402i −0.699963 + 0.404124i
\(826\) 0 0
\(827\) 239.825 + 658.914i 0.289994 + 0.796752i 0.996066 + 0.0886122i \(0.0282432\pi\)
−0.706072 + 0.708140i \(0.749535\pi\)
\(828\) 0 0
\(829\) 533.805 + 308.192i 0.643914 + 0.371764i 0.786121 0.618073i \(-0.212086\pi\)
−0.142207 + 0.989837i \(0.545420\pi\)
\(830\) 0 0
\(831\) 159.358 + 189.915i 0.191767 + 0.228538i
\(832\) 0 0
\(833\) −211.591 1199.99i −0.254011 1.44057i
\(834\) 0 0
\(835\) 1675.45i 2.00653i
\(836\) 0 0
\(837\) −757.680 −0.905233
\(838\) 0 0
\(839\) 298.301 52.5985i 0.355543 0.0626918i 0.00697601 0.999976i \(-0.497779\pi\)
0.348567 + 0.937284i \(0.386668\pi\)
\(840\) 0 0
\(841\) −727.571 + 610.504i −0.865126 + 0.725927i
\(842\) 0 0
\(843\) 187.350 324.499i 0.222242 0.384934i
\(844\) 0 0
\(845\) 1224.16 445.557i 1.44871 0.527287i
\(846\) 0 0
\(847\) 41.6565 + 72.1511i 0.0491812 + 0.0851843i
\(848\) 0 0
\(849\) −1052.92 185.659i −1.24019 0.218680i
\(850\) 0 0
\(851\) −436.286 + 519.945i −0.512674 + 0.610982i
\(852\) 0 0
\(853\) −523.403 190.503i −0.613602 0.223333i 0.0164765 0.999864i \(-0.494755\pi\)
−0.630079 + 0.776531i \(0.716977\pi\)
\(854\) 0 0
\(855\) 370.586 696.481i 0.433434 0.814597i
\(856\) 0 0
\(857\) 411.030 1129.30i 0.479615 1.31773i −0.430205 0.902731i \(-0.641559\pi\)
0.909821 0.415002i \(-0.136219\pi\)
\(858\) 0 0
\(859\) −991.552 832.011i −1.15431 0.968581i −0.154498 0.987993i \(-0.549376\pi\)
−0.999812 + 0.0194123i \(0.993820\pi\)
\(860\) 0 0
\(861\) −21.2264 + 120.381i −0.0246532 + 0.139816i
\(862\) 0 0
\(863\) 40.8465 23.5827i 0.0473308 0.0273265i −0.476148 0.879365i \(-0.657967\pi\)
0.523479 + 0.852039i \(0.324634\pi\)
\(864\) 0 0
\(865\) 207.830 + 571.007i 0.240265 + 0.660124i
\(866\) 0 0
\(867\) −665.066 383.976i −0.767089 0.442879i
\(868\) 0 0
\(869\) 690.286 + 822.651i 0.794346 + 0.946664i
\(870\) 0 0
\(871\) −51.1265 289.953i −0.0586987 0.332897i
\(872\) 0 0
\(873\) 421.208i 0.482484i
\(874\) 0 0
\(875\) −204.038 −0.233187
\(876\) 0 0
\(877\) −1675.13 + 295.370i −1.91006 + 0.336796i −0.997409 0.0719344i \(-0.977083\pi\)
−0.912655 + 0.408730i \(0.865972\pi\)
\(878\) 0 0
\(879\) −345.678 + 290.058i −0.393263 + 0.329986i
\(880\) 0 0
\(881\) −164.580 + 285.060i −0.186810 + 0.323565i −0.944185 0.329416i \(-0.893148\pi\)
0.757375 + 0.652980i \(0.226482\pi\)
\(882\) 0 0
\(883\) −705.777 + 256.882i −0.799294 + 0.290919i −0.709194 0.705014i \(-0.750941\pi\)
−0.0901001 + 0.995933i \(0.528719\pi\)
\(884\) 0 0
\(885\) 914.856 + 1584.58i 1.03374 + 1.79048i
\(886\) 0 0
\(887\) 972.986 + 171.564i 1.09694 + 0.193420i 0.692695 0.721231i \(-0.256423\pi\)
0.404245 + 0.914651i \(0.367534\pi\)
\(888\) 0 0
\(889\) −63.7960 + 76.0291i −0.0717615 + 0.0855220i
\(890\) 0 0
\(891\) −72.3154 26.3207i −0.0811621 0.0295406i
\(892\) 0 0
\(893\) 248.274 100.225i 0.278023 0.112234i
\(894\) 0 0
\(895\) −637.133 + 1750.51i −0.711880 + 1.95587i
\(896\) 0 0
\(897\) −93.2748 78.2669i −0.103985 0.0872540i
\(898\) 0 0
\(899\) 199.648 1132.26i 0.222078 1.25947i
\(900\) 0 0
\(901\) 417.212 240.877i 0.463054 0.267345i
\(902\) 0 0
\(903\) 33.3968 + 91.7569i 0.0369843 + 0.101613i
\(904\) 0 0
\(905\) −2346.05 1354.49i −2.59232 1.49668i
\(906\) 0 0
\(907\) −352.939 420.617i −0.389128 0.463745i 0.535545 0.844507i \(-0.320106\pi\)
−0.924673 + 0.380762i \(0.875662\pi\)
\(908\) 0 0
\(909\) 72.6338 + 411.927i 0.0799052 + 0.453165i
\(910\) 0 0
\(911\) 1165.31i 1.27916i −0.768725 0.639580i \(-0.779108\pi\)
0.768725 0.639580i \(-0.220892\pi\)
\(912\) 0 0
\(913\) 1040.46 1.13961
\(914\) 0 0
\(915\) 109.653 19.3348i 0.119840 0.0211310i
\(916\) 0 0
\(917\) 81.4573 68.3508i 0.0888303 0.0745374i
\(918\) 0 0
\(919\) 796.263 1379.17i 0.866445 1.50073i 0.000840474 1.00000i \(-0.499732\pi\)
0.865605 0.500728i \(-0.166934\pi\)
\(920\) 0 0
\(921\) 597.413 217.440i 0.648657 0.236092i
\(922\) 0 0
\(923\) 98.7131 + 170.976i 0.106948 + 0.185240i
\(924\) 0 0
\(925\) 1432.38 + 252.567i 1.54852 + 0.273045i
\(926\) 0 0
\(927\) 326.897 389.580i 0.352639 0.420259i
\(928\) 0 0
\(929\) −563.414 205.066i −0.606473 0.220738i 0.0204862 0.999790i \(-0.493479\pi\)
−0.626959 + 0.779052i \(0.715701\pi\)
\(930\) 0 0
\(931\) 471.455 + 754.976i 0.506396 + 0.810930i
\(932\) 0 0
\(933\) −117.925 + 323.997i −0.126394 + 0.347264i
\(934\) 0 0
\(935\) 1306.92 + 1096.64i 1.39778 + 1.17288i
\(936\) 0 0
\(937\) 87.3902 495.614i 0.0932659 0.528937i −0.901999 0.431738i \(-0.857900\pi\)
0.995265 0.0971993i \(-0.0309884\pi\)
\(938\) 0 0
\(939\) −19.1184 + 11.0380i −0.0203604 + 0.0117551i
\(940\) 0 0
\(941\) 533.231 + 1465.04i 0.566664 + 1.55690i 0.809678 + 0.586874i \(0.199642\pi\)
−0.243014 + 0.970023i \(0.578136\pi\)
\(942\) 0 0
\(943\) −713.359 411.858i −0.756478 0.436753i
\(944\) 0 0
\(945\) −215.291 256.573i −0.227821 0.271506i
\(946\) 0 0
\(947\) −77.7244 440.797i −0.0820743 0.465467i −0.997950 0.0640039i \(-0.979613\pi\)
0.915875 0.401463i \(-0.131498\pi\)
\(948\) 0 0
\(949\) 200.976i 0.211777i
\(950\) 0 0
\(951\) 335.180 0.352450
\(952\) 0 0
\(953\) −571.126 + 100.705i −0.599293 + 0.105672i −0.465061 0.885278i \(-0.653968\pi\)
−0.134232 + 0.990950i \(0.542857\pi\)
\(954\) 0 0
\(955\) −1090.84 + 915.320i −1.14224 + 0.958450i
\(956\) 0 0
\(957\) 336.012 581.990i 0.351110 0.608140i
\(958\) 0 0
\(959\) 222.578 81.0116i 0.232093 0.0844751i
\(960\) 0 0
\(961\) −111.420 192.985i −0.115942 0.200817i
\(962\) 0 0
\(963\) 627.519 + 110.648i 0.651629 + 0.114900i
\(964\) 0 0
\(965\) 821.881 979.479i 0.851690 1.01500i
\(966\) 0 0
\(967\) −1304.48 474.792i −1.34900 0.490995i −0.436363 0.899771i \(-0.643734\pi\)
−0.912635 + 0.408775i \(0.865956\pi\)
\(968\) 0 0
\(969\) 969.733 + 136.576i 1.00076 + 0.140945i
\(970\) 0 0
\(971\) 330.178 907.156i 0.340039 0.934249i −0.645344 0.763892i \(-0.723286\pi\)
0.985382 0.170357i \(-0.0544920\pi\)
\(972\) 0 0
\(973\) −214.825 180.260i −0.220786 0.185262i
\(974\) 0 0
\(975\) −45.3088 + 256.959i −0.0464706 + 0.263548i
\(976\) 0 0
\(977\) 711.436 410.748i 0.728185 0.420418i −0.0895730 0.995980i \(-0.528550\pi\)
0.817758 + 0.575563i \(0.195217\pi\)
\(978\) 0 0
\(979\) 77.2432 + 212.224i 0.0789001 + 0.216776i
\(980\) 0 0
\(981\) −706.261 407.760i −0.719940 0.415658i
\(982\) 0 0
\(983\) 19.1380 + 22.8077i 0.0194689 + 0.0232022i 0.775691 0.631113i \(-0.217402\pi\)
−0.756222 + 0.654315i \(0.772957\pi\)
\(984\) 0 0
\(985\) 26.4258 + 149.868i 0.0268282 + 0.152150i
\(986\) 0 0
\(987\) 40.9754i 0.0415151i
\(988\) 0 0
\(989\) −657.996 −0.665315
\(990\) 0 0
\(991\) 880.518 155.259i 0.888514 0.156669i 0.289282 0.957244i \(-0.406584\pi\)
0.599233 + 0.800575i \(0.295472\pi\)
\(992\) 0 0
\(993\) −138.700 + 116.383i −0.139677 + 0.117203i
\(994\) 0 0
\(995\) −140.254 + 242.926i −0.140958 + 0.244147i
\(996\) 0 0
\(997\) 45.1018 16.4157i 0.0452375 0.0164651i −0.319302 0.947653i \(-0.603449\pi\)
0.364540 + 0.931188i \(0.381226\pi\)
\(998\) 0 0
\(999\) 483.006 + 836.591i 0.483489 + 0.837428i
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 76.3.j.a.53.2 yes 18
4.3 odd 2 304.3.z.b.129.2 18
19.9 even 9 1444.3.c.c.721.12 18
19.10 odd 18 1444.3.c.c.721.7 18
19.14 odd 18 inner 76.3.j.a.33.2 18
76.71 even 18 304.3.z.b.33.2 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.3.j.a.33.2 18 19.14 odd 18 inner
76.3.j.a.53.2 yes 18 1.1 even 1 trivial
304.3.z.b.33.2 18 76.71 even 18
304.3.z.b.129.2 18 4.3 odd 2
1444.3.c.c.721.7 18 19.10 odd 18
1444.3.c.c.721.12 18 19.9 even 9