Properties

Label 64.4.e.a.17.5
Level $64$
Weight $4$
Character 64.17
Analytic conductor $3.776$
Analytic rank $0$
Dimension $10$
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] = [64,4,Mod(17,64)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(64, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 64.e (of order \(4\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.77612224037\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - x^{8} + 6x^{7} + 14x^{6} - 80x^{5} + 56x^{4} + 96x^{3} - 64x^{2} - 512x + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 17.5
Root \(1.97476 + 0.316760i\) of defining polynomial
Character \(\chi\) \(=\) 64.17
Dual form 64.4.e.a.49.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(5.96513 + 5.96513i) q^{3} +(8.67959 - 8.67959i) q^{5} +1.63924i q^{7} +44.1656i q^{9} +O(q^{10})\) \(q+(5.96513 + 5.96513i) q^{3} +(8.67959 - 8.67959i) q^{5} +1.63924i q^{7} +44.1656i q^{9} +(-18.2021 + 18.2021i) q^{11} +(-9.34700 - 9.34700i) q^{13} +103.550 q^{15} +53.6113 q^{17} +(-70.9870 - 70.9870i) q^{19} +(-9.77831 + 9.77831i) q^{21} -25.1189i q^{23} -25.6706i q^{25} +(-102.395 + 102.395i) q^{27} +(-181.094 - 181.094i) q^{29} -132.684 q^{31} -217.155 q^{33} +(14.2280 + 14.2280i) q^{35} +(174.872 - 174.872i) q^{37} -111.512i q^{39} +198.660i q^{41} +(285.717 - 285.717i) q^{43} +(383.339 + 383.339i) q^{45} -78.3629 q^{47} +340.313 q^{49} +(319.799 + 319.799i) q^{51} +(-525.776 + 525.776i) q^{53} +315.973i q^{55} -846.894i q^{57} +(-46.5301 + 46.5301i) q^{59} +(193.318 + 193.318i) q^{61} -72.3982 q^{63} -162.256 q^{65} +(-282.182 - 282.182i) q^{67} +(149.838 - 149.838i) q^{69} +727.536i q^{71} +106.065i q^{73} +(153.128 - 153.128i) q^{75} +(-29.8376 - 29.8376i) q^{77} +58.9970 q^{79} -29.1298 q^{81} +(410.156 + 410.156i) q^{83} +(465.324 - 465.324i) q^{85} -2160.50i q^{87} +768.959i q^{89} +(15.3220 - 15.3220i) q^{91} +(-791.477 - 791.477i) q^{93} -1232.28 q^{95} -809.953 q^{97} +(-803.905 - 803.905i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{3} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{3} - 2 q^{5} - 18 q^{11} - 2 q^{13} + 124 q^{15} - 4 q^{17} + 26 q^{19} + 52 q^{21} - 184 q^{27} - 202 q^{29} - 368 q^{31} - 4 q^{33} - 476 q^{35} - 10 q^{37} + 838 q^{43} + 194 q^{45} + 944 q^{47} + 94 q^{49} + 1500 q^{51} - 378 q^{53} - 1706 q^{59} + 910 q^{61} - 2628 q^{63} - 492 q^{65} - 1942 q^{67} + 580 q^{69} + 2954 q^{75} - 268 q^{77} + 4416 q^{79} + 482 q^{81} + 2562 q^{83} - 12 q^{85} - 3332 q^{91} - 2192 q^{93} - 6900 q^{95} - 4 q^{97} - 4958 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/64\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(63\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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
\(3\) 5.96513 + 5.96513i 1.14799 + 1.14799i 0.986947 + 0.161043i \(0.0514857\pi\)
0.161043 + 0.986947i \(0.448514\pi\)
\(4\) 0 0
\(5\) 8.67959 8.67959i 0.776326 0.776326i −0.202878 0.979204i \(-0.565029\pi\)
0.979204 + 0.202878i \(0.0650295\pi\)
\(6\) 0 0
\(7\) 1.63924i 0.0885109i 0.999020 + 0.0442554i \(0.0140915\pi\)
−0.999020 + 0.0442554i \(0.985908\pi\)
\(8\) 0 0
\(9\) 44.1656i 1.63576i
\(10\) 0 0
\(11\) −18.2021 + 18.2021i −0.498921 + 0.498921i −0.911102 0.412181i \(-0.864767\pi\)
0.412181 + 0.911102i \(0.364767\pi\)
\(12\) 0 0
\(13\) −9.34700 9.34700i −0.199415 0.199415i 0.600334 0.799749i \(-0.295034\pi\)
−0.799749 + 0.600334i \(0.795034\pi\)
\(14\) 0 0
\(15\) 103.550 1.78243
\(16\) 0 0
\(17\) 53.6113 0.764862 0.382431 0.923984i \(-0.375087\pi\)
0.382431 + 0.923984i \(0.375087\pi\)
\(18\) 0 0
\(19\) −70.9870 70.9870i −0.857133 0.857133i 0.133866 0.990999i \(-0.457261\pi\)
−0.990999 + 0.133866i \(0.957261\pi\)
\(20\) 0 0
\(21\) −9.77831 + 9.77831i −0.101610 + 0.101610i
\(22\) 0 0
\(23\) 25.1189i 0.227724i −0.993497 0.113862i \(-0.963678\pi\)
0.993497 0.113862i \(-0.0363222\pi\)
\(24\) 0 0
\(25\) 25.6706i 0.205365i
\(26\) 0 0
\(27\) −102.395 + 102.395i −0.729850 + 0.729850i
\(28\) 0 0
\(29\) −181.094 181.094i −1.15960 1.15960i −0.984563 0.175033i \(-0.943997\pi\)
−0.175033 0.984563i \(-0.556003\pi\)
\(30\) 0 0
\(31\) −132.684 −0.768733 −0.384367 0.923180i \(-0.625580\pi\)
−0.384367 + 0.923180i \(0.625580\pi\)
\(32\) 0 0
\(33\) −217.155 −1.14551
\(34\) 0 0
\(35\) 14.2280 + 14.2280i 0.0687133 + 0.0687133i
\(36\) 0 0
\(37\) 174.872 174.872i 0.776994 0.776994i −0.202324 0.979319i \(-0.564849\pi\)
0.979319 + 0.202324i \(0.0648495\pi\)
\(38\) 0 0
\(39\) 111.512i 0.457853i
\(40\) 0 0
\(41\) 198.660i 0.756720i 0.925658 + 0.378360i \(0.123512\pi\)
−0.925658 + 0.378360i \(0.876488\pi\)
\(42\) 0 0
\(43\) 285.717 285.717i 1.01329 1.01329i 0.0133770 0.999911i \(-0.495742\pi\)
0.999911 0.0133770i \(-0.00425815\pi\)
\(44\) 0 0
\(45\) 383.339 + 383.339i 1.26989 + 1.26989i
\(46\) 0 0
\(47\) −78.3629 −0.243200 −0.121600 0.992579i \(-0.538803\pi\)
−0.121600 + 0.992579i \(0.538803\pi\)
\(48\) 0 0
\(49\) 340.313 0.992166
\(50\) 0 0
\(51\) 319.799 + 319.799i 0.878054 + 0.878054i
\(52\) 0 0
\(53\) −525.776 + 525.776i −1.36266 + 1.36266i −0.492143 + 0.870515i \(0.663786\pi\)
−0.870515 + 0.492143i \(0.836214\pi\)
\(54\) 0 0
\(55\) 315.973i 0.774650i
\(56\) 0 0
\(57\) 846.894i 1.96796i
\(58\) 0 0
\(59\) −46.5301 + 46.5301i −0.102673 + 0.102673i −0.756577 0.653904i \(-0.773130\pi\)
0.653904 + 0.756577i \(0.273130\pi\)
\(60\) 0 0
\(61\) 193.318 + 193.318i 0.405767 + 0.405767i 0.880259 0.474493i \(-0.157368\pi\)
−0.474493 + 0.880259i \(0.657368\pi\)
\(62\) 0 0
\(63\) −72.3982 −0.144783
\(64\) 0 0
\(65\) −162.256 −0.309622
\(66\) 0 0
\(67\) −282.182 282.182i −0.514538 0.514538i 0.401375 0.915914i \(-0.368532\pi\)
−0.915914 + 0.401375i \(0.868532\pi\)
\(68\) 0 0
\(69\) 149.838 149.838i 0.261425 0.261425i
\(70\) 0 0
\(71\) 727.536i 1.21609i 0.793901 + 0.608046i \(0.208047\pi\)
−0.793901 + 0.608046i \(0.791953\pi\)
\(72\) 0 0
\(73\) 106.065i 0.170054i 0.996379 + 0.0850270i \(0.0270977\pi\)
−0.996379 + 0.0850270i \(0.972902\pi\)
\(74\) 0 0
\(75\) 153.128 153.128i 0.235757 0.235757i
\(76\) 0 0
\(77\) −29.8376 29.8376i −0.0441599 0.0441599i
\(78\) 0 0
\(79\) 58.9970 0.0840213 0.0420107 0.999117i \(-0.486624\pi\)
0.0420107 + 0.999117i \(0.486624\pi\)
\(80\) 0 0
\(81\) −29.1298 −0.0399586
\(82\) 0 0
\(83\) 410.156 + 410.156i 0.542416 + 0.542416i 0.924236 0.381821i \(-0.124703\pi\)
−0.381821 + 0.924236i \(0.624703\pi\)
\(84\) 0 0
\(85\) 465.324 465.324i 0.593783 0.593783i
\(86\) 0 0
\(87\) 2160.50i 2.66241i
\(88\) 0 0
\(89\) 768.959i 0.915837i 0.888994 + 0.457918i \(0.151405\pi\)
−0.888994 + 0.457918i \(0.848595\pi\)
\(90\) 0 0
\(91\) 15.3220 15.3220i 0.0176504 0.0176504i
\(92\) 0 0
\(93\) −791.477 791.477i −0.882499 0.882499i
\(94\) 0 0
\(95\) −1232.28 −1.33083
\(96\) 0 0
\(97\) −809.953 −0.847817 −0.423908 0.905705i \(-0.639342\pi\)
−0.423908 + 0.905705i \(0.639342\pi\)
\(98\) 0 0
\(99\) −803.905 803.905i −0.816116 0.816116i
\(100\) 0 0
\(101\) −303.189 + 303.189i −0.298698 + 0.298698i −0.840504 0.541806i \(-0.817741\pi\)
0.541806 + 0.840504i \(0.317741\pi\)
\(102\) 0 0
\(103\) 962.201i 0.920471i 0.887797 + 0.460235i \(0.152235\pi\)
−0.887797 + 0.460235i \(0.847765\pi\)
\(104\) 0 0
\(105\) 169.743i 0.157764i
\(106\) 0 0
\(107\) −728.337 + 728.337i −0.658046 + 0.658046i −0.954918 0.296871i \(-0.904057\pi\)
0.296871 + 0.954918i \(0.404057\pi\)
\(108\) 0 0
\(109\) −593.258 593.258i −0.521319 0.521319i 0.396651 0.917970i \(-0.370172\pi\)
−0.917970 + 0.396651i \(0.870172\pi\)
\(110\) 0 0
\(111\) 2086.27 1.78396
\(112\) 0 0
\(113\) 351.938 0.292987 0.146493 0.989212i \(-0.453201\pi\)
0.146493 + 0.989212i \(0.453201\pi\)
\(114\) 0 0
\(115\) −218.022 218.022i −0.176788 0.176788i
\(116\) 0 0
\(117\) 412.816 412.816i 0.326195 0.326195i
\(118\) 0 0
\(119\) 87.8821i 0.0676986i
\(120\) 0 0
\(121\) 668.370i 0.502157i
\(122\) 0 0
\(123\) −1185.04 + 1185.04i −0.868707 + 0.868707i
\(124\) 0 0
\(125\) 862.139 + 862.139i 0.616896 + 0.616896i
\(126\) 0 0
\(127\) 2365.81 1.65301 0.826504 0.562931i \(-0.190326\pi\)
0.826504 + 0.562931i \(0.190326\pi\)
\(128\) 0 0
\(129\) 3408.67 2.32649
\(130\) 0 0
\(131\) −403.454 403.454i −0.269083 0.269083i 0.559647 0.828731i \(-0.310937\pi\)
−0.828731 + 0.559647i \(0.810937\pi\)
\(132\) 0 0
\(133\) 116.365 116.365i 0.0758656 0.0758656i
\(134\) 0 0
\(135\) 1777.50i 1.13320i
\(136\) 0 0
\(137\) 856.850i 0.534348i −0.963648 0.267174i \(-0.913910\pi\)
0.963648 0.267174i \(-0.0860898\pi\)
\(138\) 0 0
\(139\) 1689.33 1689.33i 1.03084 1.03084i 0.0313345 0.999509i \(-0.490024\pi\)
0.999509 0.0313345i \(-0.00997572\pi\)
\(140\) 0 0
\(141\) −467.445 467.445i −0.279191 0.279191i
\(142\) 0 0
\(143\) 340.269 0.198984
\(144\) 0 0
\(145\) −3143.64 −1.80045
\(146\) 0 0
\(147\) 2030.01 + 2030.01i 1.13900 + 1.13900i
\(148\) 0 0
\(149\) −32.2208 + 32.2208i −0.0177156 + 0.0177156i −0.715909 0.698193i \(-0.753988\pi\)
0.698193 + 0.715909i \(0.253988\pi\)
\(150\) 0 0
\(151\) 1077.06i 0.580460i −0.956957 0.290230i \(-0.906268\pi\)
0.956957 0.290230i \(-0.0937319\pi\)
\(152\) 0 0
\(153\) 2367.78i 1.25113i
\(154\) 0 0
\(155\) −1151.64 + 1151.64i −0.596788 + 0.596788i
\(156\) 0 0
\(157\) 1905.61 + 1905.61i 0.968691 + 0.968691i 0.999525 0.0308332i \(-0.00981607\pi\)
−0.0308332 + 0.999525i \(0.509816\pi\)
\(158\) 0 0
\(159\) −6272.64 −3.12863
\(160\) 0 0
\(161\) 41.1761 0.0201561
\(162\) 0 0
\(163\) −709.828 709.828i −0.341092 0.341092i 0.515686 0.856778i \(-0.327537\pi\)
−0.856778 + 0.515686i \(0.827537\pi\)
\(164\) 0 0
\(165\) −1884.82 + 1884.82i −0.889291 + 0.889291i
\(166\) 0 0
\(167\) 3460.66i 1.60356i −0.597623 0.801778i \(-0.703888\pi\)
0.597623 0.801778i \(-0.296112\pi\)
\(168\) 0 0
\(169\) 2022.27i 0.920467i
\(170\) 0 0
\(171\) 3135.18 3135.18i 1.40207 1.40207i
\(172\) 0 0
\(173\) −1303.54 1303.54i −0.572871 0.572871i 0.360059 0.932930i \(-0.382757\pi\)
−0.932930 + 0.360059i \(0.882757\pi\)
\(174\) 0 0
\(175\) 42.0803 0.0181770
\(176\) 0 0
\(177\) −555.117 −0.235735
\(178\) 0 0
\(179\) 1082.35 + 1082.35i 0.451947 + 0.451947i 0.896000 0.444054i \(-0.146460\pi\)
−0.444054 + 0.896000i \(0.646460\pi\)
\(180\) 0 0
\(181\) 2943.93 2943.93i 1.20895 1.20895i 0.237588 0.971366i \(-0.423643\pi\)
0.971366 0.237588i \(-0.0763567\pi\)
\(182\) 0 0
\(183\) 2306.33i 0.931633i
\(184\) 0 0
\(185\) 3035.64i 1.20640i
\(186\) 0 0
\(187\) −975.837 + 975.837i −0.381606 + 0.381606i
\(188\) 0 0
\(189\) −167.851 167.851i −0.0645997 0.0645997i
\(190\) 0 0
\(191\) 430.650 0.163145 0.0815726 0.996667i \(-0.474006\pi\)
0.0815726 + 0.996667i \(0.474006\pi\)
\(192\) 0 0
\(193\) −2266.98 −0.845497 −0.422749 0.906247i \(-0.638935\pi\)
−0.422749 + 0.906247i \(0.638935\pi\)
\(194\) 0 0
\(195\) −967.881 967.881i −0.355443 0.355443i
\(196\) 0 0
\(197\) −1039.41 + 1039.41i −0.375913 + 0.375913i −0.869625 0.493712i \(-0.835640\pi\)
0.493712 + 0.869625i \(0.335640\pi\)
\(198\) 0 0
\(199\) 4989.44i 1.77735i 0.458540 + 0.888674i \(0.348373\pi\)
−0.458540 + 0.888674i \(0.651627\pi\)
\(200\) 0 0
\(201\) 3366.51i 1.18137i
\(202\) 0 0
\(203\) 296.857 296.857i 0.102637 0.102637i
\(204\) 0 0
\(205\) 1724.29 + 1724.29i 0.587462 + 0.587462i
\(206\) 0 0
\(207\) 1109.39 0.372503
\(208\) 0 0
\(209\) 2584.22 0.855283
\(210\) 0 0
\(211\) −2651.50 2651.50i −0.865103 0.865103i 0.126823 0.991925i \(-0.459522\pi\)
−0.991925 + 0.126823i \(0.959522\pi\)
\(212\) 0 0
\(213\) −4339.85 + 4339.85i −1.39606 + 1.39606i
\(214\) 0 0
\(215\) 4959.80i 1.57328i
\(216\) 0 0
\(217\) 217.501i 0.0680413i
\(218\) 0 0
\(219\) −632.691 + 632.691i −0.195220 + 0.195220i
\(220\) 0 0
\(221\) −501.105 501.105i −0.152525 0.152525i
\(222\) 0 0
\(223\) −3690.85 −1.10833 −0.554165 0.832407i \(-0.686962\pi\)
−0.554165 + 0.832407i \(0.686962\pi\)
\(224\) 0 0
\(225\) 1133.76 0.335928
\(226\) 0 0
\(227\) 1710.42 + 1710.42i 0.500108 + 0.500108i 0.911471 0.411363i \(-0.134947\pi\)
−0.411363 + 0.911471i \(0.634947\pi\)
\(228\) 0 0
\(229\) −91.8012 + 91.8012i −0.0264908 + 0.0264908i −0.720228 0.693737i \(-0.755963\pi\)
0.693737 + 0.720228i \(0.255963\pi\)
\(230\) 0 0
\(231\) 355.971i 0.101390i
\(232\) 0 0
\(233\) 4259.71i 1.19769i 0.800863 + 0.598847i \(0.204374\pi\)
−0.800863 + 0.598847i \(0.795626\pi\)
\(234\) 0 0
\(235\) −680.158 + 680.158i −0.188803 + 0.188803i
\(236\) 0 0
\(237\) 351.925 + 351.925i 0.0964556 + 0.0964556i
\(238\) 0 0
\(239\) 5053.12 1.36761 0.683806 0.729664i \(-0.260324\pi\)
0.683806 + 0.729664i \(0.260324\pi\)
\(240\) 0 0
\(241\) 48.8379 0.0130536 0.00652681 0.999979i \(-0.497922\pi\)
0.00652681 + 0.999979i \(0.497922\pi\)
\(242\) 0 0
\(243\) 2590.91 + 2590.91i 0.683978 + 0.683978i
\(244\) 0 0
\(245\) 2953.78 2953.78i 0.770244 0.770244i
\(246\) 0 0
\(247\) 1327.03i 0.341850i
\(248\) 0 0
\(249\) 4893.28i 1.24538i
\(250\) 0 0
\(251\) 2604.76 2604.76i 0.655025 0.655025i −0.299174 0.954199i \(-0.596711\pi\)
0.954199 + 0.299174i \(0.0967111\pi\)
\(252\) 0 0
\(253\) 457.216 + 457.216i 0.113616 + 0.113616i
\(254\) 0 0
\(255\) 5551.44 1.36331
\(256\) 0 0
\(257\) 739.054 0.179381 0.0896905 0.995970i \(-0.471412\pi\)
0.0896905 + 0.995970i \(0.471412\pi\)
\(258\) 0 0
\(259\) 286.658 + 286.658i 0.0687724 + 0.0687724i
\(260\) 0 0
\(261\) 7998.12 7998.12i 1.89682 1.89682i
\(262\) 0 0
\(263\) 2448.30i 0.574025i 0.957927 + 0.287012i \(0.0926621\pi\)
−0.957927 + 0.287012i \(0.907338\pi\)
\(264\) 0 0
\(265\) 9127.03i 2.11573i
\(266\) 0 0
\(267\) −4586.94 + 4586.94i −1.05137 + 1.05137i
\(268\) 0 0
\(269\) −829.952 829.952i −0.188116 0.188116i 0.606765 0.794881i \(-0.292467\pi\)
−0.794881 + 0.606765i \(0.792467\pi\)
\(270\) 0 0
\(271\) −1404.85 −0.314902 −0.157451 0.987527i \(-0.550328\pi\)
−0.157451 + 0.987527i \(0.550328\pi\)
\(272\) 0 0
\(273\) 182.796 0.0405249
\(274\) 0 0
\(275\) 467.257 + 467.257i 0.102461 + 0.102461i
\(276\) 0 0
\(277\) −2245.69 + 2245.69i −0.487112 + 0.487112i −0.907394 0.420281i \(-0.861931\pi\)
0.420281 + 0.907394i \(0.361931\pi\)
\(278\) 0 0
\(279\) 5860.07i 1.25747i
\(280\) 0 0
\(281\) 6045.97i 1.28353i −0.766900 0.641766i \(-0.778202\pi\)
0.766900 0.641766i \(-0.221798\pi\)
\(282\) 0 0
\(283\) −2459.63 + 2459.63i −0.516643 + 0.516643i −0.916554 0.399911i \(-0.869041\pi\)
0.399911 + 0.916554i \(0.369041\pi\)
\(284\) 0 0
\(285\) −7350.69 7350.69i −1.52778 1.52778i
\(286\) 0 0
\(287\) −325.653 −0.0669780
\(288\) 0 0
\(289\) −2038.82 −0.414986
\(290\) 0 0
\(291\) −4831.48 4831.48i −0.973286 0.973286i
\(292\) 0 0
\(293\) −1852.49 + 1852.49i −0.369364 + 0.369364i −0.867245 0.497881i \(-0.834112\pi\)
0.497881 + 0.867245i \(0.334112\pi\)
\(294\) 0 0
\(295\) 807.725i 0.159415i
\(296\) 0 0
\(297\) 3727.60i 0.728275i
\(298\) 0 0
\(299\) −234.787 + 234.787i −0.0454116 + 0.0454116i
\(300\) 0 0
\(301\) 468.359 + 468.359i 0.0896870 + 0.0896870i
\(302\) 0 0
\(303\) −3617.13 −0.685804
\(304\) 0 0
\(305\) 3355.83 0.630015
\(306\) 0 0
\(307\) −2107.35 2107.35i −0.391768 0.391768i 0.483549 0.875317i \(-0.339347\pi\)
−0.875317 + 0.483549i \(0.839347\pi\)
\(308\) 0 0
\(309\) −5739.66 + 5739.66i −1.05669 + 1.05669i
\(310\) 0 0
\(311\) 5294.90i 0.965422i 0.875780 + 0.482711i \(0.160348\pi\)
−0.875780 + 0.482711i \(0.839652\pi\)
\(312\) 0 0
\(313\) 4005.87i 0.723403i −0.932294 0.361702i \(-0.882196\pi\)
0.932294 0.361702i \(-0.117804\pi\)
\(314\) 0 0
\(315\) −628.387 + 628.387i −0.112399 + 0.112399i
\(316\) 0 0
\(317\) 809.240 + 809.240i 0.143380 + 0.143380i 0.775153 0.631773i \(-0.217673\pi\)
−0.631773 + 0.775153i \(0.717673\pi\)
\(318\) 0 0
\(319\) 6592.56 1.15709
\(320\) 0 0
\(321\) −8689.25 −1.51086
\(322\) 0 0
\(323\) −3805.71 3805.71i −0.655589 0.655589i
\(324\) 0 0
\(325\) −239.943 + 239.943i −0.0409527 + 0.0409527i
\(326\) 0 0
\(327\) 7077.72i 1.19694i
\(328\) 0 0
\(329\) 128.456i 0.0215259i
\(330\) 0 0
\(331\) 4229.66 4229.66i 0.702366 0.702366i −0.262552 0.964918i \(-0.584564\pi\)
0.964918 + 0.262552i \(0.0845641\pi\)
\(332\) 0 0
\(333\) 7723.33 + 7723.33i 1.27098 + 1.27098i
\(334\) 0 0
\(335\) −4898.46 −0.798899
\(336\) 0 0
\(337\) 10002.6 1.61684 0.808419 0.588607i \(-0.200323\pi\)
0.808419 + 0.588607i \(0.200323\pi\)
\(338\) 0 0
\(339\) 2099.36 + 2099.36i 0.336346 + 0.336346i
\(340\) 0 0
\(341\) 2415.12 2415.12i 0.383537 0.383537i
\(342\) 0 0
\(343\) 1120.12i 0.176328i
\(344\) 0 0
\(345\) 2601.06i 0.405903i
\(346\) 0 0
\(347\) 6409.49 6409.49i 0.991583 0.991583i −0.00838198 0.999965i \(-0.502668\pi\)
0.999965 + 0.00838198i \(0.00266810\pi\)
\(348\) 0 0
\(349\) −5503.23 5503.23i −0.844071 0.844071i 0.145314 0.989386i \(-0.453581\pi\)
−0.989386 + 0.145314i \(0.953581\pi\)
\(350\) 0 0
\(351\) 1914.18 0.291086
\(352\) 0 0
\(353\) −1411.35 −0.212800 −0.106400 0.994323i \(-0.533932\pi\)
−0.106400 + 0.994323i \(0.533932\pi\)
\(354\) 0 0
\(355\) 6314.71 + 6314.71i 0.944085 + 0.944085i
\(356\) 0 0
\(357\) −524.228 + 524.228i −0.0777174 + 0.0777174i
\(358\) 0 0
\(359\) 2160.73i 0.317658i −0.987306 0.158829i \(-0.949228\pi\)
0.987306 0.158829i \(-0.0507718\pi\)
\(360\) 0 0
\(361\) 3219.31i 0.469355i
\(362\) 0 0
\(363\) −3986.92 + 3986.92i −0.576471 + 0.576471i
\(364\) 0 0
\(365\) 920.599 + 920.599i 0.132017 + 0.132017i
\(366\) 0 0
\(367\) −10757.7 −1.53010 −0.765052 0.643969i \(-0.777287\pi\)
−0.765052 + 0.643969i \(0.777287\pi\)
\(368\) 0 0
\(369\) −8773.95 −1.23782
\(370\) 0 0
\(371\) −861.875 861.875i −0.120610 0.120610i
\(372\) 0 0
\(373\) 1406.99 1406.99i 0.195312 0.195312i −0.602675 0.797987i \(-0.705898\pi\)
0.797987 + 0.602675i \(0.205898\pi\)
\(374\) 0 0
\(375\) 10285.5i 1.41638i
\(376\) 0 0
\(377\) 3385.37i 0.462481i
\(378\) 0 0
\(379\) 1146.95 1146.95i 0.155449 0.155449i −0.625098 0.780547i \(-0.714941\pi\)
0.780547 + 0.625098i \(0.214941\pi\)
\(380\) 0 0
\(381\) 14112.4 + 14112.4i 1.89764 + 1.89764i
\(382\) 0 0
\(383\) −9042.17 −1.20635 −0.603176 0.797608i \(-0.706098\pi\)
−0.603176 + 0.797608i \(0.706098\pi\)
\(384\) 0 0
\(385\) −517.957 −0.0685650
\(386\) 0 0
\(387\) 12618.8 + 12618.8i 1.65750 + 1.65750i
\(388\) 0 0
\(389\) 2575.34 2575.34i 0.335668 0.335668i −0.519066 0.854734i \(-0.673720\pi\)
0.854734 + 0.519066i \(0.173720\pi\)
\(390\) 0 0
\(391\) 1346.66i 0.174178i
\(392\) 0 0
\(393\) 4813.31i 0.617810i
\(394\) 0 0
\(395\) 512.070 512.070i 0.0652279 0.0652279i
\(396\) 0 0
\(397\) −7121.46 7121.46i −0.900292 0.900292i 0.0951695 0.995461i \(-0.469661\pi\)
−0.995461 + 0.0951695i \(0.969661\pi\)
\(398\) 0 0
\(399\) 1388.27 0.174186
\(400\) 0 0
\(401\) −3025.14 −0.376729 −0.188365 0.982099i \(-0.560319\pi\)
−0.188365 + 0.982099i \(0.560319\pi\)
\(402\) 0 0
\(403\) 1240.20 + 1240.20i 0.153297 + 0.153297i
\(404\) 0 0
\(405\) −252.835 + 252.835i −0.0310209 + 0.0310209i
\(406\) 0 0
\(407\) 6366.06i 0.775317i
\(408\) 0 0
\(409\) 9440.21i 1.14129i −0.821196 0.570646i \(-0.806693\pi\)
0.821196 0.570646i \(-0.193307\pi\)
\(410\) 0 0
\(411\) 5111.22 5111.22i 0.613426 0.613426i
\(412\) 0 0
\(413\) −76.2743 76.2743i −0.00908768 0.00908768i
\(414\) 0 0
\(415\) 7119.98 0.842183
\(416\) 0 0
\(417\) 20154.2 2.36680
\(418\) 0 0
\(419\) 3255.69 + 3255.69i 0.379597 + 0.379597i 0.870957 0.491360i \(-0.163500\pi\)
−0.491360 + 0.870957i \(0.663500\pi\)
\(420\) 0 0
\(421\) −9438.04 + 9438.04i −1.09259 + 1.09259i −0.0973423 + 0.995251i \(0.531034\pi\)
−0.995251 + 0.0973423i \(0.968966\pi\)
\(422\) 0 0
\(423\) 3460.95i 0.397818i
\(424\) 0 0
\(425\) 1376.23i 0.157076i
\(426\) 0 0
\(427\) −316.895 + 316.895i −0.0359148 + 0.0359148i
\(428\) 0 0
\(429\) 2029.75 + 2029.75i 0.228432 + 0.228432i
\(430\) 0 0
\(431\) 10617.7 1.18663 0.593314 0.804971i \(-0.297819\pi\)
0.593314 + 0.804971i \(0.297819\pi\)
\(432\) 0 0
\(433\) 706.479 0.0784093 0.0392046 0.999231i \(-0.487518\pi\)
0.0392046 + 0.999231i \(0.487518\pi\)
\(434\) 0 0
\(435\) −18752.2 18752.2i −2.06690 2.06690i
\(436\) 0 0
\(437\) −1783.12 + 1783.12i −0.195190 + 0.195190i
\(438\) 0 0
\(439\) 13611.8i 1.47985i −0.672688 0.739926i \(-0.734860\pi\)
0.672688 0.739926i \(-0.265140\pi\)
\(440\) 0 0
\(441\) 15030.1i 1.62295i
\(442\) 0 0
\(443\) −3126.97 + 3126.97i −0.335366 + 0.335366i −0.854620 0.519254i \(-0.826210\pi\)
0.519254 + 0.854620i \(0.326210\pi\)
\(444\) 0 0
\(445\) 6674.25 + 6674.25i 0.710988 + 0.710988i
\(446\) 0 0
\(447\) −384.403 −0.0406748
\(448\) 0 0
\(449\) −5231.76 −0.549893 −0.274947 0.961460i \(-0.588660\pi\)
−0.274947 + 0.961460i \(0.588660\pi\)
\(450\) 0 0
\(451\) −3616.03 3616.03i −0.377543 0.377543i
\(452\) 0 0
\(453\) 6424.78 6424.78i 0.666363 0.666363i
\(454\) 0 0
\(455\) 265.978i 0.0274049i
\(456\) 0 0
\(457\) 6833.10i 0.699429i −0.936856 0.349715i \(-0.886279\pi\)
0.936856 0.349715i \(-0.113721\pi\)
\(458\) 0 0
\(459\) −5489.54 + 5489.54i −0.558235 + 0.558235i
\(460\) 0 0
\(461\) −5975.90 5975.90i −0.603742 0.603742i 0.337561 0.941304i \(-0.390398\pi\)
−0.941304 + 0.337561i \(0.890398\pi\)
\(462\) 0 0
\(463\) 4273.38 0.428943 0.214472 0.976730i \(-0.431197\pi\)
0.214472 + 0.976730i \(0.431197\pi\)
\(464\) 0 0
\(465\) −13739.4 −1.37021
\(466\) 0 0
\(467\) −12245.6 12245.6i −1.21340 1.21340i −0.969901 0.243500i \(-0.921704\pi\)
−0.243500 0.969901i \(-0.578296\pi\)
\(468\) 0 0
\(469\) 462.566 462.566i 0.0455422 0.0455422i
\(470\) 0 0
\(471\) 22734.5i 2.22410i
\(472\) 0 0
\(473\) 10401.3i 1.01110i
\(474\) 0 0
\(475\) −1822.28 + 1822.28i −0.176025 + 0.176025i
\(476\) 0 0
\(477\) −23221.2 23221.2i −2.22898 2.22898i
\(478\) 0 0
\(479\) 4067.97 0.388038 0.194019 0.980998i \(-0.437848\pi\)
0.194019 + 0.980998i \(0.437848\pi\)
\(480\) 0 0
\(481\) −3269.06 −0.309888
\(482\) 0 0
\(483\) 245.621 + 245.621i 0.0231390 + 0.0231390i
\(484\) 0 0
\(485\) −7030.06 + 7030.06i −0.658182 + 0.658182i
\(486\) 0 0
\(487\) 16174.3i 1.50499i −0.658600 0.752493i \(-0.728851\pi\)
0.658600 0.752493i \(-0.271149\pi\)
\(488\) 0 0
\(489\) 8468.44i 0.783141i
\(490\) 0 0
\(491\) −13596.7 + 13596.7i −1.24971 + 1.24971i −0.293866 + 0.955847i \(0.594942\pi\)
−0.955847 + 0.293866i \(0.905058\pi\)
\(492\) 0 0
\(493\) −9708.68 9708.68i −0.886931 0.886931i
\(494\) 0 0
\(495\) −13955.1 −1.26714
\(496\) 0 0
\(497\) −1192.61 −0.107637
\(498\) 0 0
\(499\) 14646.7 + 14646.7i 1.31398 + 1.31398i 0.918453 + 0.395530i \(0.129439\pi\)
0.395530 + 0.918453i \(0.370561\pi\)
\(500\) 0 0
\(501\) 20643.3 20643.3i 1.84087 1.84087i
\(502\) 0 0
\(503\) 9828.84i 0.871265i −0.900125 0.435632i \(-0.856525\pi\)
0.900125 0.435632i \(-0.143475\pi\)
\(504\) 0 0
\(505\) 5263.12i 0.463774i
\(506\) 0 0
\(507\) 12063.1 12063.1i 1.05669 1.05669i
\(508\) 0 0
\(509\) 13456.1 + 13456.1i 1.17177 + 1.17177i 0.981787 + 0.189985i \(0.0608439\pi\)
0.189985 + 0.981787i \(0.439156\pi\)
\(510\) 0 0
\(511\) −173.866 −0.0150516
\(512\) 0 0
\(513\) 14537.5 1.25116
\(514\) 0 0
\(515\) 8351.51 + 8351.51i 0.714585 + 0.714585i
\(516\) 0 0
\(517\) 1426.37 1426.37i 0.121338 0.121338i
\(518\) 0 0
\(519\) 15551.6i 1.31530i
\(520\) 0 0
\(521\) 10607.1i 0.891950i 0.895045 + 0.445975i \(0.147143\pi\)
−0.895045 + 0.445975i \(0.852857\pi\)
\(522\) 0 0
\(523\) −3903.15 + 3903.15i −0.326334 + 0.326334i −0.851191 0.524857i \(-0.824119\pi\)
0.524857 + 0.851191i \(0.324119\pi\)
\(524\) 0 0
\(525\) 251.015 + 251.015i 0.0208670 + 0.0208670i
\(526\) 0 0
\(527\) −7113.36 −0.587975
\(528\) 0 0
\(529\) 11536.0 0.948142
\(530\) 0 0
\(531\) −2055.03 2055.03i −0.167949 0.167949i
\(532\) 0 0
\(533\) 1856.88 1856.88i 0.150901 0.150901i
\(534\) 0 0
\(535\) 12643.3i 1.02172i
\(536\) 0 0
\(537\) 12912.7i 1.03766i
\(538\) 0 0
\(539\) −6194.39 + 6194.39i −0.495012 + 0.495012i
\(540\) 0 0
\(541\) 9532.77 + 9532.77i 0.757570 + 0.757570i 0.975880 0.218309i \(-0.0700541\pi\)
−0.218309 + 0.975880i \(0.570054\pi\)
\(542\) 0 0
\(543\) 35121.9 2.77573
\(544\) 0 0
\(545\) −10298.5 −0.809427
\(546\) 0 0
\(547\) 1232.88 + 1232.88i 0.0963693 + 0.0963693i 0.753648 0.657278i \(-0.228292\pi\)
−0.657278 + 0.753648i \(0.728292\pi\)
\(548\) 0 0
\(549\) −8537.99 + 8537.99i −0.663739 + 0.663739i
\(550\) 0 0
\(551\) 25710.6i 1.98786i
\(552\) 0 0
\(553\) 96.7105i 0.00743680i
\(554\) 0 0
\(555\) 18108.0 18108.0i 1.38494 1.38494i
\(556\) 0 0
\(557\) −2889.57 2889.57i −0.219812 0.219812i 0.588607 0.808419i \(-0.299676\pi\)
−0.808419 + 0.588607i \(0.799676\pi\)
\(558\) 0 0
\(559\) −5341.19 −0.404129
\(560\) 0 0
\(561\) −11642.0 −0.876159
\(562\) 0 0
\(563\) −70.0753 70.0753i −0.00524569 0.00524569i 0.704479 0.709725i \(-0.251181\pi\)
−0.709725 + 0.704479i \(0.751181\pi\)
\(564\) 0 0
\(565\) 3054.68 3054.68i 0.227453 0.227453i
\(566\) 0 0
\(567\) 47.7509i 0.00353677i
\(568\) 0 0
\(569\) 8915.23i 0.656847i −0.944531 0.328423i \(-0.893483\pi\)
0.944531 0.328423i \(-0.106517\pi\)
\(570\) 0 0
\(571\) −4946.30 + 4946.30i −0.362515 + 0.362515i −0.864738 0.502223i \(-0.832516\pi\)
0.502223 + 0.864738i \(0.332516\pi\)
\(572\) 0 0
\(573\) 2568.89 + 2568.89i 0.187289 + 0.187289i
\(574\) 0 0
\(575\) −644.818 −0.0467665
\(576\) 0 0
\(577\) 17911.5 1.29232 0.646159 0.763203i \(-0.276374\pi\)
0.646159 + 0.763203i \(0.276374\pi\)
\(578\) 0 0
\(579\) −13522.8 13522.8i −0.970622 0.970622i
\(580\) 0 0
\(581\) −672.347 + 672.347i −0.0480097 + 0.0480097i
\(582\) 0 0
\(583\) 19140.4i 1.35972i
\(584\) 0 0
\(585\) 7166.15i 0.506468i
\(586\) 0 0
\(587\) −7940.26 + 7940.26i −0.558312 + 0.558312i −0.928827 0.370514i \(-0.879181\pi\)
0.370514 + 0.928827i \(0.379181\pi\)
\(588\) 0 0
\(589\) 9418.83 + 9418.83i 0.658907 + 0.658907i
\(590\) 0 0
\(591\) −12400.4 −0.863088
\(592\) 0 0
\(593\) −7006.26 −0.485181 −0.242591 0.970129i \(-0.577997\pi\)
−0.242591 + 0.970129i \(0.577997\pi\)
\(594\) 0 0
\(595\) 762.780 + 762.780i 0.0525562 + 0.0525562i
\(596\) 0 0
\(597\) −29762.7 + 29762.7i −2.04038 + 2.04038i
\(598\) 0 0
\(599\) 8502.74i 0.579987i 0.957029 + 0.289994i \(0.0936532\pi\)
−0.957029 + 0.289994i \(0.906347\pi\)
\(600\) 0 0
\(601\) 11936.2i 0.810127i −0.914289 0.405063i \(-0.867249\pi\)
0.914289 0.405063i \(-0.132751\pi\)
\(602\) 0 0
\(603\) 12462.8 12462.8i 0.841663 0.841663i
\(604\) 0 0
\(605\) 5801.18 + 5801.18i 0.389837 + 0.389837i
\(606\) 0 0
\(607\) 3850.00 0.257441 0.128721 0.991681i \(-0.458913\pi\)
0.128721 + 0.991681i \(0.458913\pi\)
\(608\) 0 0
\(609\) 3541.58 0.235652
\(610\) 0 0
\(611\) 732.459 + 732.459i 0.0484977 + 0.0484977i
\(612\) 0 0
\(613\) −6320.36 + 6320.36i −0.416439 + 0.416439i −0.883974 0.467536i \(-0.845142\pi\)
0.467536 + 0.883974i \(0.345142\pi\)
\(614\) 0 0
\(615\) 20571.2i 1.34880i
\(616\) 0 0
\(617\) 2585.09i 0.168674i −0.996437 0.0843370i \(-0.973123\pi\)
0.996437 0.0843370i \(-0.0268772\pi\)
\(618\) 0 0
\(619\) −7325.02 + 7325.02i −0.475634 + 0.475634i −0.903732 0.428098i \(-0.859184\pi\)
0.428098 + 0.903732i \(0.359184\pi\)
\(620\) 0 0
\(621\) 2572.06 + 2572.06i 0.166205 + 0.166205i
\(622\) 0 0
\(623\) −1260.51 −0.0810615
\(624\) 0 0
\(625\) 18174.8 1.16319
\(626\) 0 0
\(627\) 15415.2 + 15415.2i 0.981857 + 0.981857i
\(628\) 0 0
\(629\) 9375.13 9375.13i 0.594294 0.594294i
\(630\) 0 0
\(631\) 14411.5i 0.909210i −0.890693 0.454605i \(-0.849780\pi\)
0.890693 0.454605i \(-0.150220\pi\)
\(632\) 0 0
\(633\) 31633.1i 1.98626i
\(634\) 0 0
\(635\) 20534.3 20534.3i 1.28327 1.28327i
\(636\) 0 0
\(637\) −3180.91 3180.91i −0.197853 0.197853i
\(638\) 0 0
\(639\) −32132.1 −1.98924
\(640\) 0 0
\(641\) −25724.0 −1.58508 −0.792542 0.609818i \(-0.791243\pi\)
−0.792542 + 0.609818i \(0.791243\pi\)
\(642\) 0 0
\(643\) 7835.74 + 7835.74i 0.480578 + 0.480578i 0.905316 0.424738i \(-0.139634\pi\)
−0.424738 + 0.905316i \(0.639634\pi\)
\(644\) 0 0
\(645\) 29585.9 29585.9i 1.80611 1.80611i
\(646\) 0 0
\(647\) 1247.43i 0.0757981i 0.999282 + 0.0378991i \(0.0120665\pi\)
−0.999282 + 0.0378991i \(0.987933\pi\)
\(648\) 0 0
\(649\) 1693.89i 0.102451i
\(650\) 0 0
\(651\) 1297.42 1297.42i 0.0781107 0.0781107i
\(652\) 0 0
\(653\) 8302.21 + 8302.21i 0.497535 + 0.497535i 0.910670 0.413135i \(-0.135566\pi\)
−0.413135 + 0.910670i \(0.635566\pi\)
\(654\) 0 0
\(655\) −7003.63 −0.417793
\(656\) 0 0
\(657\) −4684.42 −0.278168
\(658\) 0 0
\(659\) −1696.16 1696.16i −0.100262 0.100262i 0.655196 0.755459i \(-0.272586\pi\)
−0.755459 + 0.655196i \(0.772586\pi\)
\(660\) 0 0
\(661\) −8788.30 + 8788.30i −0.517134 + 0.517134i −0.916703 0.399569i \(-0.869160\pi\)
0.399569 + 0.916703i \(0.369160\pi\)
\(662\) 0 0
\(663\) 5978.32i 0.350194i
\(664\) 0 0
\(665\) 2020.00i 0.117793i
\(666\) 0 0
\(667\) −4548.88 + 4548.88i −0.264068 + 0.264068i
\(668\) 0 0
\(669\) −22016.4 22016.4i −1.27235 1.27235i
\(670\) 0 0
\(671\) −7037.56 −0.404891
\(672\) 0 0
\(673\) −23869.3 −1.36716 −0.683578 0.729878i \(-0.739577\pi\)
−0.683578 + 0.729878i \(0.739577\pi\)
\(674\) 0 0
\(675\) 2628.54 + 2628.54i 0.149885 + 0.149885i
\(676\) 0 0
\(677\) −5663.30 + 5663.30i −0.321504 + 0.321504i −0.849344 0.527840i \(-0.823002\pi\)
0.527840 + 0.849344i \(0.323002\pi\)
\(678\) 0 0
\(679\) 1327.71i 0.0750410i
\(680\) 0 0
\(681\) 20405.8i 1.14824i
\(682\) 0 0
\(683\) 9152.80 9152.80i 0.512770 0.512770i −0.402604 0.915374i \(-0.631895\pi\)
0.915374 + 0.402604i \(0.131895\pi\)
\(684\) 0 0
\(685\) −7437.11 7437.11i −0.414828 0.414828i
\(686\) 0 0
\(687\) −1095.21 −0.0608224
\(688\) 0 0
\(689\) 9828.85 0.543468
\(690\) 0 0
\(691\) −17057.9 17057.9i −0.939091 0.939091i 0.0591580 0.998249i \(-0.481158\pi\)
−0.998249 + 0.0591580i \(0.981158\pi\)
\(692\) 0 0
\(693\) 1317.80 1317.80i 0.0722351 0.0722351i
\(694\) 0 0
\(695\) 29325.4i 1.60054i
\(696\) 0 0
\(697\) 10650.4i 0.578787i
\(698\) 0 0
\(699\) −25409.7 + 25409.7i −1.37494 + 1.37494i
\(700\) 0 0
\(701\) −7720.44 7720.44i −0.415973 0.415973i 0.467840 0.883813i \(-0.345032\pi\)
−0.883813 + 0.467840i \(0.845032\pi\)
\(702\) 0 0
\(703\) −24827.3 −1.33198
\(704\) 0 0
\(705\) −8114.47 −0.433487
\(706\) 0 0
\(707\) −497.001 497.001i −0.0264380 0.0264380i
\(708\) 0 0
\(709\) 4577.66 4577.66i 0.242479 0.242479i −0.575396 0.817875i \(-0.695152\pi\)
0.817875 + 0.575396i \(0.195152\pi\)
\(710\) 0 0
\(711\) 2605.64i 0.137439i
\(712\) 0 0
\(713\) 3332.88i 0.175059i
\(714\) 0 0
\(715\) 2953.40 2953.40i 0.154477 0.154477i
\(716\) 0 0
\(717\) 30142.5 + 30142.5i 1.57000 + 1.57000i
\(718\) 0 0
\(719\) 30210.0 1.56696 0.783479 0.621418i \(-0.213443\pi\)
0.783479 + 0.621418i \(0.213443\pi\)
\(720\) 0 0
\(721\) −1577.28 −0.0814717
\(722\) 0 0
\(723\) 291.324 + 291.324i 0.0149854 + 0.0149854i
\(724\) 0 0
\(725\) −4648.78 + 4648.78i −0.238140 + 0.238140i
\(726\) 0 0
\(727\) 20721.3i 1.05710i 0.848903 + 0.528549i \(0.177264\pi\)
−0.848903 + 0.528549i \(0.822736\pi\)
\(728\) 0 0
\(729\) 31696.7i 1.61036i
\(730\) 0 0
\(731\) 15317.6 15317.6i 0.775025 0.775025i
\(732\) 0 0
\(733\) −13879.8 13879.8i −0.699404 0.699404i 0.264878 0.964282i \(-0.414668\pi\)
−0.964282 + 0.264878i \(0.914668\pi\)
\(734\) 0 0
\(735\) 35239.3 1.76847
\(736\) 0 0
\(737\) 10272.6 0.513428
\(738\) 0 0
\(739\) −8793.93 8793.93i −0.437740 0.437740i 0.453511 0.891251i \(-0.350171\pi\)
−0.891251 + 0.453511i \(0.850171\pi\)
\(740\) 0 0
\(741\) −7915.92 + 7915.92i −0.392441 + 0.392441i
\(742\) 0 0
\(743\) 7669.27i 0.378678i 0.981912 + 0.189339i \(0.0606346\pi\)
−0.981912 + 0.189339i \(0.939365\pi\)
\(744\) 0 0
\(745\) 559.327i 0.0275062i
\(746\) 0 0
\(747\) −18114.8 + 18114.8i −0.887264 + 0.887264i
\(748\) 0 0
\(749\) −1193.92 1193.92i −0.0582443 0.0582443i
\(750\) 0 0
\(751\) 26531.8 1.28916 0.644580 0.764537i \(-0.277032\pi\)
0.644580 + 0.764537i \(0.277032\pi\)
\(752\) 0 0
\(753\) 31075.5 1.50392
\(754\) 0 0
\(755\) −9348.40 9348.40i −0.450627 0.450627i
\(756\) 0 0
\(757\) −79.4192 + 79.4192i −0.00381313 + 0.00381313i −0.709011 0.705198i \(-0.750858\pi\)
0.705198 + 0.709011i \(0.250858\pi\)
\(758\) 0 0
\(759\) 5454.71i 0.260861i
\(760\) 0 0
\(761\) 36991.3i 1.76207i −0.473055 0.881033i \(-0.656849\pi\)
0.473055 0.881033i \(-0.343151\pi\)
\(762\) 0 0
\(763\) 972.494 972.494i 0.0461424 0.0461424i
\(764\) 0 0
\(765\) 20551.3 + 20551.3i 0.971288 + 0.971288i
\(766\) 0 0
\(767\) 869.835 0.0409490
\(768\) 0 0
\(769\) 26637.0 1.24910 0.624548 0.780987i \(-0.285283\pi\)
0.624548 + 0.780987i \(0.285283\pi\)
\(770\) 0 0
\(771\) 4408.55 + 4408.55i 0.205928 + 0.205928i
\(772\) 0 0
\(773\) 19743.6 19743.6i 0.918667 0.918667i −0.0782657 0.996933i \(-0.524938\pi\)
0.996933 + 0.0782657i \(0.0249382\pi\)
\(774\) 0 0
\(775\) 3406.07i 0.157871i
\(776\) 0 0
\(777\) 3419.91i 0.157900i
\(778\) 0 0
\(779\) 14102.3 14102.3i 0.648610 0.648610i
\(780\) 0 0
\(781\) −13242.6 13242.6i −0.606734 0.606734i
\(782\) 0 0
\(783\) 37086.2 1.69266
\(784\) 0 0
\(785\) 33079.9 1.50404
\(786\) 0 0
\(787\) 28878.1 + 28878.1i 1.30800 + 1.30800i 0.922860 + 0.385136i \(0.125845\pi\)
0.385136 + 0.922860i \(0.374155\pi\)
\(788\) 0 0
\(789\) −14604.4 + 14604.4i −0.658975 + 0.658975i
\(790\) 0 0
\(791\) 576.912i 0.0259325i
\(792\) 0 0
\(793\) 3613.88i 0.161832i
\(794\) 0 0
\(795\) −54444.0 + 54444.0i −2.42884 + 2.42884i
\(796\) 0 0
\(797\) −16656.0 16656.0i −0.740257 0.740257i 0.232371 0.972627i \(-0.425352\pi\)
−0.972627 + 0.232371i \(0.925352\pi\)
\(798\) 0 0
\(799\) −4201.14 −0.186015
\(800\) 0 0
\(801\) −33961.5 −1.49809
\(802\) 0 0
\(803\) −1930.60 1930.60i −0.0848435 0.0848435i
\(804\) 0 0
\(805\) 357.391 357.391i 0.0156477 0.0156477i
\(806\) 0 0
\(807\) 9901.55i 0.431910i
\(808\) 0 0
\(809\) 34940.4i 1.51847i 0.650819 + 0.759233i \(0.274426\pi\)
−0.650819 + 0.759233i \(0.725574\pi\)
\(810\) 0 0
\(811\) 15168.2 15168.2i 0.656753 0.656753i −0.297857 0.954610i \(-0.596272\pi\)
0.954610 + 0.297857i \(0.0962719\pi\)
\(812\) 0 0
\(813\) −8380.10 8380.10i −0.361504 0.361504i
\(814\) 0 0
\(815\) −12322.0 −0.529597
\(816\) 0 0
\(817\) −40564.3 −1.73705
\(818\) 0 0
\(819\) 676.707 + 676.707i 0.0288718 + 0.0288718i
\(820\) 0 0
\(821\) 8710.55 8710.55i 0.370280 0.370280i −0.497299 0.867579i \(-0.665675\pi\)
0.867579 + 0.497299i \(0.165675\pi\)
\(822\) 0 0
\(823\) 24493.5i 1.03741i 0.854952 + 0.518707i \(0.173586\pi\)
−0.854952 + 0.518707i \(0.826414\pi\)
\(824\) 0 0
\(825\) 5574.50i 0.235248i
\(826\) 0 0
\(827\) −26328.0 + 26328.0i −1.10703 + 1.10703i −0.113492 + 0.993539i \(0.536204\pi\)
−0.993539 + 0.113492i \(0.963796\pi\)
\(828\) 0 0
\(829\) 9108.25 + 9108.25i 0.381596 + 0.381596i 0.871677 0.490081i \(-0.163033\pi\)
−0.490081 + 0.871677i \(0.663033\pi\)
\(830\) 0 0
\(831\) −26791.6 −1.11840
\(832\) 0 0
\(833\) 18244.6 0.758870
\(834\) 0 0
\(835\) −30037.1 30037.1i −1.24488 1.24488i
\(836\) 0 0
\(837\) 13586.2 13586.2i 0.561060 0.561060i
\(838\) 0 0
\(839\) 1394.89i 0.0573982i 0.999588 + 0.0286991i \(0.00913646\pi\)
−0.999588 + 0.0286991i \(0.990864\pi\)
\(840\) 0 0
\(841\) 41200.9i 1.68932i
\(842\) 0 0
\(843\) 36065.0 36065.0i 1.47348 1.47348i
\(844\) 0 0
\(845\) −17552.4 17552.4i −0.714583 0.714583i
\(846\) 0 0
\(847\) −1095.62 −0.0444463
\(848\) 0 0
\(849\) −29344.1 −1.18620
\(850\) 0 0
\(851\) −4392.60 4392.60i −0.176941 0.176941i
\(852\) 0 0
\(853\) 15284.7 15284.7i 0.613527 0.613527i −0.330337 0.943863i \(-0.607162\pi\)
0.943863 + 0.330337i \(0.107162\pi\)
\(854\) 0 0
\(855\) 54424.2i 2.17692i
\(856\) 0 0
\(857\) 2273.70i 0.0906277i 0.998973 + 0.0453139i \(0.0144288\pi\)
−0.998973 + 0.0453139i \(0.985571\pi\)
\(858\) 0 0
\(859\) 21674.3 21674.3i 0.860905 0.860905i −0.130538 0.991443i \(-0.541671\pi\)
0.991443 + 0.130538i \(0.0416706\pi\)
\(860\) 0 0
\(861\) −1942.56 1942.56i −0.0768900 0.0768900i
\(862\) 0 0
\(863\) −23721.7 −0.935686 −0.467843 0.883812i \(-0.654969\pi\)
−0.467843 + 0.883812i \(0.654969\pi\)
\(864\) 0 0
\(865\) −22628.5 −0.889469
\(866\) 0 0
\(867\) −12161.9 12161.9i −0.476400 0.476400i
\(868\) 0 0
\(869\) −1073.87 + 1073.87i −0.0419200 + 0.0419200i
\(870\) 0 0
\(871\) 5275.12i 0.205213i
\(872\) 0 0
\(873\) 35772.1i 1.38683i
\(874\) 0 0
\(875\) −1413.26 + 1413.26i −0.0546020 + 0.0546020i
\(876\) 0 0
\(877\) 22429.6 + 22429.6i 0.863617 + 0.863617i 0.991756 0.128139i \(-0.0409004\pi\)
−0.128139 + 0.991756i \(0.540900\pi\)
\(878\) 0 0
\(879\) −22100.7 −0.848054
\(880\) 0 0
\(881\) 24603.0 0.940859 0.470429 0.882438i \(-0.344099\pi\)
0.470429 + 0.882438i \(0.344099\pi\)
\(882\) 0 0
\(883\) 23486.7 + 23486.7i 0.895120 + 0.895120i 0.995000 0.0998799i \(-0.0318459\pi\)
−0.0998799 + 0.995000i \(0.531846\pi\)
\(884\) 0 0
\(885\) −4818.19 + 4818.19i −0.183007 + 0.183007i
\(886\) 0 0
\(887\) 39722.9i 1.50368i −0.659345 0.751841i \(-0.729166\pi\)
0.659345 0.751841i \(-0.270834\pi\)
\(888\) 0 0
\(889\) 3878.15i 0.146309i
\(890\) 0 0
\(891\) 530.222 530.222i 0.0199362 0.0199362i
\(892\) 0 0
\(893\) 5562.75 + 5562.75i 0.208455 + 0.208455i
\(894\) 0 0
\(895\) 18788.7 0.701716
\(896\) 0 0
\(897\) −2801.07 −0.104264
\(898\) 0 0
\(899\) 24028.2 + 24028.2i 0.891420 + 0.891420i
\(900\) 0 0
\(901\) −28187.5 + 28187.5i −1.04225 + 1.04225i
\(902\) 0 0
\(903\) 5587.65i 0.205920i
\(904\) 0 0
\(905\) 51104.2i 1.87708i
\(906\) 0 0
\(907\) −4565.44 + 4565.44i −0.167136 + 0.167136i −0.785719 0.618583i \(-0.787707\pi\)
0.618583 + 0.785719i \(0.287707\pi\)
\(908\) 0 0
\(909\) −13390.5 13390.5i −0.488599 0.488599i
\(910\) 0 0
\(911\) 2013.95 0.0732438 0.0366219 0.999329i \(-0.488340\pi\)
0.0366219 + 0.999329i \(0.488340\pi\)
\(912\) 0 0
\(913\) −14931.4 −0.541245
\(914\) 0 0
\(915\) 20018.0 + 20018.0i 0.723251 + 0.723251i
\(916\) 0 0
\(917\) 661.359 661.359i 0.0238168 0.0238168i
\(918\) 0 0
\(919\) 37746.5i 1.35489i 0.735575 + 0.677443i \(0.236912\pi\)
−0.735575 + 0.677443i \(0.763088\pi\)
\(920\) 0 0
\(921\) 25141.2i 0.899492i
\(922\) 0 0
\(923\) 6800.28 6800.28i 0.242507 0.242507i
\(924\) 0 0
\(925\) −4489.07 4489.07i −0.159567 0.159567i
\(926\) 0 0
\(927\) −42496.2 −1.50567
\(928\) 0 0
\(929\) 45643.5 1.61197 0.805983 0.591939i \(-0.201637\pi\)
0.805983 + 0.591939i \(0.201637\pi\)
\(930\) 0 0
\(931\) −24157.8 24157.8i −0.850418 0.850418i
\(932\) 0 0
\(933\) −31584.8 + 31584.8i −1.10829 + 1.10829i
\(934\) 0 0
\(935\) 16939.7i 0.592501i
\(936\) 0 0
\(937\) 47317.5i 1.64973i −0.565331 0.824864i \(-0.691252\pi\)
0.565331 0.824864i \(-0.308748\pi\)
\(938\) 0 0
\(939\) 23895.6 23895.6i 0.830460 0.830460i
\(940\) 0 0
\(941\) 15074.7 + 15074.7i 0.522234 + 0.522234i 0.918246 0.396012i \(-0.129606\pi\)
−0.396012 + 0.918246i \(0.629606\pi\)
\(942\) 0 0
\(943\) 4990.14 0.172324
\(944\) 0 0
\(945\) −2913.75 −0.100301
\(946\) 0 0
\(947\) −14567.7 14567.7i −0.499880 0.499880i 0.411521 0.911400i \(-0.364998\pi\)
−0.911400 + 0.411521i \(0.864998\pi\)
\(948\) 0 0
\(949\) 991.388 991.388i 0.0339113 0.0339113i
\(950\) 0 0
\(951\) 9654.45i 0.329198i
\(952\) 0 0
\(953\) 42987.2i 1.46117i 0.682824 + 0.730583i \(0.260752\pi\)
−0.682824 + 0.730583i \(0.739248\pi\)
\(954\) 0 0
\(955\) 3737.87 3737.87i 0.126654 0.126654i
\(956\) 0 0
\(957\) 39325.5 + 39325.5i 1.32833 + 1.32833i
\(958\) 0 0
\(959\) 1404.59 0.0472956
\(960\) 0 0
\(961\) −12186.0 −0.409049
\(962\) 0 0
\(963\) −32167.4 32167.4i −1.07641 1.07641i
\(964\) 0 0
\(965\) −19676.5 + 19676.5i −0.656381 + 0.656381i
\(966\) 0 0
\(967\) 44030.7i 1.46425i 0.681170 + 0.732126i \(0.261472\pi\)
−0.681170 + 0.732126i \(0.738528\pi\)
\(968\) 0 0
\(969\) 45403.1i 1.50522i
\(970\) 0 0
\(971\) −35699.8 + 35699.8i −1.17988 + 1.17988i −0.200101 + 0.979775i \(0.564127\pi\)
−0.979775 + 0.200101i \(0.935873\pi\)
\(972\) 0 0
\(973\) 2769.23 + 2769.23i 0.0912409 + 0.0912409i
\(974\) 0 0
\(975\) −2862.58 −0.0940267
\(976\) 0 0
\(977\) −49515.3 −1.62143 −0.810714 0.585442i \(-0.800921\pi\)
−0.810714 + 0.585442i \(0.800921\pi\)
\(978\) 0 0
\(979\) −13996.6 13996.6i −0.456930 0.456930i
\(980\) 0 0
\(981\) 26201.6 26201.6i 0.852755 0.852755i
\(982\) 0 0
\(983\) 40046.2i 1.29936i −0.760206 0.649682i \(-0.774902\pi\)
0.760206 0.649682i \(-0.225098\pi\)
\(984\) 0 0
\(985\) 18043.3i 0.583662i
\(986\) 0 0
\(987\) 766.257 766.257i 0.0247115 0.0247115i
\(988\) 0 0
\(989\) −7176.90 7176.90i −0.230750 0.230750i
\(990\) 0 0
\(991\) −18673.2 −0.598560 −0.299280 0.954165i \(-0.596746\pi\)
−0.299280 + 0.954165i \(0.596746\pi\)
\(992\) 0 0
\(993\) 50460.9 1.61262
\(994\) 0 0
\(995\) 43306.3 + 43306.3i 1.37980 + 1.37980i
\(996\) 0 0
\(997\) 21982.1 21982.1i 0.698274 0.698274i −0.265764 0.964038i \(-0.585624\pi\)
0.964038 + 0.265764i \(0.0856243\pi\)
\(998\) 0 0
\(999\) 35812.1i 1.13418i
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 64.4.e.a.17.5 10
3.2 odd 2 576.4.k.a.145.1 10
4.3 odd 2 16.4.e.a.13.2 yes 10
8.3 odd 2 128.4.e.b.33.5 10
8.5 even 2 128.4.e.a.33.1 10
12.11 even 2 144.4.k.a.109.4 10
16.3 odd 4 128.4.e.b.97.5 10
16.5 even 4 inner 64.4.e.a.49.5 10
16.11 odd 4 16.4.e.a.5.2 10
16.13 even 4 128.4.e.a.97.1 10
32.3 odd 8 1024.4.b.j.513.1 10
32.5 even 8 1024.4.a.m.1.1 10
32.11 odd 8 1024.4.a.n.1.1 10
32.13 even 8 1024.4.b.k.513.1 10
32.19 odd 8 1024.4.b.j.513.10 10
32.21 even 8 1024.4.a.m.1.10 10
32.27 odd 8 1024.4.a.n.1.10 10
32.29 even 8 1024.4.b.k.513.10 10
48.5 odd 4 576.4.k.a.433.1 10
48.11 even 4 144.4.k.a.37.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.4.e.a.5.2 10 16.11 odd 4
16.4.e.a.13.2 yes 10 4.3 odd 2
64.4.e.a.17.5 10 1.1 even 1 trivial
64.4.e.a.49.5 10 16.5 even 4 inner
128.4.e.a.33.1 10 8.5 even 2
128.4.e.a.97.1 10 16.13 even 4
128.4.e.b.33.5 10 8.3 odd 2
128.4.e.b.97.5 10 16.3 odd 4
144.4.k.a.37.4 10 48.11 even 4
144.4.k.a.109.4 10 12.11 even 2
576.4.k.a.145.1 10 3.2 odd 2
576.4.k.a.433.1 10 48.5 odd 4
1024.4.a.m.1.1 10 32.5 even 8
1024.4.a.m.1.10 10 32.21 even 8
1024.4.a.n.1.1 10 32.11 odd 8
1024.4.a.n.1.10 10 32.27 odd 8
1024.4.b.j.513.1 10 32.3 odd 8
1024.4.b.j.513.10 10 32.19 odd 8
1024.4.b.k.513.1 10 32.13 even 8
1024.4.b.k.513.10 10 32.29 even 8