Properties

Label 128.4.e.b.33.5
Level $128$
Weight $4$
Character 128.33
Analytic conductor $7.552$
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] = [128,4,Mod(33,128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(128, 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("128.33");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 128.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: \(7.55224448073\)
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 33.5
Root \(1.97476 + 0.316760i\) of defining polynomial
Character \(\chi\) \(=\) 128.33
Dual form 128.4.e.b.97.5

$q$-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 \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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 −0.979204 0.202878i \(-0.934971\pi\)
0.202878 + 0.979204i \(0.434971\pi\)
\(6\) 0 0
\(7\) 1.63924i 0.0885109i −0.999020 0.0442554i \(-0.985908\pi\)
0.999020 0.0442554i \(-0.0140915\pi\)
\(8\) 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 0.799749 0.600334i \(-0.204966\pi\)
−0.600334 + 0.799749i \(0.704966\pi\)
\(14\) 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 \(0.0363222\pi\)
−0.993497 + 0.113862i \(0.963678\pi\)
\(24\) 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 \(0.0560031\pi\)
0.175033 + 0.984563i \(0.443997\pi\)
\(30\) 0 0
\(31\) 132.684 0.768733 0.384367 0.923180i \(-0.374420\pi\)
0.384367 + 0.923180i \(0.374420\pi\)
\(32\) 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 −0.979319 0.202324i \(-0.935151\pi\)
0.202324 + 0.979319i \(0.435151\pi\)
\(38\) 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 \(-0.461197\pi\)
0.121600 + 0.992579i \(0.461197\pi\)
\(48\) 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 \(-0.336214\pi\)
0.870515 0.492143i \(-0.163786\pi\)
\(54\) 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 0.474493 0.880259i \(-0.342632\pi\)
−0.880259 + 0.474493i \(0.842632\pi\)
\(62\) 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 \(-0.791953\pi\)
0.793901 0.608046i \(-0.208047\pi\)
\(72\) 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 \(-0.513376\pi\)
−0.0420107 + 0.999117i \(0.513376\pi\)
\(80\) 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 −0.541806 0.840504i \(-0.682259\pi\)
0.840504 + 0.541806i \(0.182259\pi\)
\(102\) 0 0
\(103\) 962.201i 0.920471i −0.887797 0.460235i \(-0.847765\pi\)
0.887797 0.460235i \(-0.152235\pi\)
\(104\) 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 0.917970 0.396651i \(-0.129828\pi\)
−0.396651 + 0.917970i \(0.629828\pi\)
\(110\) 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 \(-0.809674\pi\)
−0.826504 + 0.562931i \(0.809674\pi\)
\(128\) 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 −0.698193 0.715909i \(-0.746012\pi\)
0.715909 + 0.698193i \(0.246012\pi\)
\(150\) 0 0
\(151\) 1077.06i 0.580460i 0.956957 + 0.290230i \(0.0937319\pi\)
−0.956957 + 0.290230i \(0.906268\pi\)
\(152\) 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 0.0308332 0.999525i \(-0.490184\pi\)
−0.999525 + 0.0308332i \(0.990184\pi\)
\(158\) 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 \(0.296112\pi\)
−0.597623 + 0.801778i \(0.703888\pi\)
\(168\) 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 0.932930 0.360059i \(-0.117243\pi\)
−0.360059 + 0.932930i \(0.617243\pi\)
\(174\) 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 \(0.576357\pi\)
−0.971366 + 0.237588i \(0.923643\pi\)
\(182\) 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 \(-0.525994\pi\)
−0.0815726 + 0.996667i \(0.525994\pi\)
\(192\) 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 −0.493712 0.869625i \(-0.664360\pi\)
0.869625 + 0.493712i \(0.164360\pi\)
\(198\) 0 0
\(199\) 4989.44i 1.77735i −0.458540 0.888674i \(-0.651627\pi\)
0.458540 0.888674i \(-0.348373\pi\)
\(200\) 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 \(-0.313038\pi\)
0.554165 + 0.832407i \(0.313038\pi\)
\(224\) 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 −0.693737 0.720228i \(-0.744037\pi\)
0.720228 + 0.693737i \(0.244037\pi\)
\(230\) 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 \(-0.739676\pi\)
−0.683806 + 0.729664i \(0.739676\pi\)
\(240\) 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 \(-0.907338\pi\)
0.957927 0.287012i \(-0.0926621\pi\)
\(264\) 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 0.794881 0.606765i \(-0.207533\pi\)
−0.606765 + 0.794881i \(0.707533\pi\)
\(270\) 0 0
\(271\) 1404.85 0.314902 0.157451 0.987527i \(-0.449672\pi\)
0.157451 + 0.987527i \(0.449672\pi\)
\(272\) 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 −0.420281 0.907394i \(-0.638069\pi\)
0.907394 + 0.420281i \(0.138069\pi\)
\(278\) 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 −0.497881 0.867245i \(-0.665888\pi\)
0.867245 + 0.497881i \(0.165888\pi\)
\(294\) 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 \(-0.839652\pi\)
0.875780 0.482711i \(-0.160348\pi\)
\(312\) 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 0.631773 0.775153i \(-0.282327\pi\)
−0.775153 + 0.631773i \(0.782327\pi\)
\(318\) 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 0.989386 0.145314i \(-0.0464193\pi\)
−0.145314 + 0.989386i \(0.546419\pi\)
\(350\) 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 \(0.0507718\pi\)
−0.987306 + 0.158829i \(0.949228\pi\)
\(360\) 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 \(-0.222713\pi\)
0.765052 + 0.643969i \(0.222713\pi\)
\(368\) 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 −0.797987 0.602675i \(-0.794102\pi\)
0.602675 + 0.797987i \(0.294102\pi\)
\(374\) 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 \(-0.293902\pi\)
0.603176 + 0.797608i \(0.293902\pi\)
\(384\) 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 −0.854734 0.519066i \(-0.826280\pi\)
0.519066 + 0.854734i \(0.326280\pi\)
\(390\) 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 0.995461 0.0951695i \(-0.0303393\pi\)
−0.0951695 + 0.995461i \(0.530339\pi\)
\(398\) 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 \(-0.468966\pi\)
0.995251 0.0973423i \(-0.0310342\pi\)
\(422\) 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 \(-0.702181\pi\)
−0.593314 + 0.804971i \(0.702181\pi\)
\(432\) 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 \(0.265140\pi\)
−0.672688 + 0.739926i \(0.734860\pi\)
\(440\) 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 0.941304 0.337561i \(-0.109602\pi\)
−0.337561 + 0.941304i \(0.609602\pi\)
\(462\) 0 0
\(463\) −4273.38 −0.428943 −0.214472 0.976730i \(-0.568803\pi\)
−0.214472 + 0.976730i \(0.568803\pi\)
\(464\) 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 \(-0.562152\pi\)
−0.194019 + 0.980998i \(0.562152\pi\)
\(480\) 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 \(0.271149\pi\)
−0.658600 + 0.752493i \(0.728851\pi\)
\(488\) 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 \(0.143475\pi\)
−0.900125 + 0.435632i \(0.856525\pi\)
\(504\) 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 \(-0.939156\pi\)
−0.189985 0.981787i \(-0.560844\pi\)
\(510\) 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 0.218309 0.975880i \(-0.429946\pi\)
−0.975880 + 0.218309i \(0.929946\pi\)
\(542\) 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 0.808419 0.588607i \(-0.200324\pi\)
−0.588607 + 0.808419i \(0.700324\pi\)
\(558\) 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 \(-0.906347\pi\)
0.957029 0.289994i \(-0.0936532\pi\)
\(600\) 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 \(-0.541087\pi\)
−0.128721 + 0.991681i \(0.541087\pi\)
\(608\) 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 −0.467536 0.883974i \(-0.654858\pi\)
0.883974 + 0.467536i \(0.154858\pi\)
\(614\) 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 \(0.150220\pi\)
−0.890693 + 0.454605i \(0.849780\pi\)
\(632\) 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 \(-0.987933\pi\)
0.999282 0.0378991i \(-0.0120665\pi\)
\(648\) 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 0.413135 0.910670i \(-0.364434\pi\)
−0.910670 + 0.413135i \(0.864434\pi\)
\(654\) 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 −0.399569 0.916703i \(-0.630840\pi\)
0.916703 + 0.399569i \(0.130840\pi\)
\(662\) 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 −0.527840 0.849344i \(-0.676998\pi\)
0.849344 + 0.527840i \(0.176998\pi\)
\(678\) 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 0.883813 0.467840i \(-0.154968\pi\)
−0.467840 + 0.883813i \(0.654968\pi\)
\(702\) 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 −0.817875 0.575396i \(-0.804848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(710\) 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 \(-0.786557\pi\)
−0.783479 + 0.621418i \(0.786557\pi\)
\(720\) 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 \(-0.822736\pi\)
0.848903 0.528549i \(-0.177264\pi\)
\(728\) 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 0.964282 0.264878i \(-0.0853316\pi\)
−0.264878 + 0.964282i \(0.585332\pi\)
\(734\) 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 \(-0.939365\pi\)
0.981912 0.189339i \(-0.0606346\pi\)
\(744\) 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 \(-0.722968\pi\)
−0.644580 + 0.764537i \(0.722968\pi\)
\(752\) 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 −0.705198 0.709011i \(-0.749142\pi\)
0.709011 + 0.705198i \(0.249142\pi\)
\(758\) 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 −0.996933 0.0782657i \(-0.975062\pi\)
0.0782657 + 0.996933i \(0.475062\pi\)
\(774\) 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 0.972627 0.232371i \(-0.0746483\pi\)
−0.232371 + 0.972627i \(0.574648\pi\)
\(798\) 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 −0.867579 0.497299i \(-0.834325\pi\)
0.497299 + 0.867579i \(0.334325\pi\)
\(822\) 0 0
\(823\) 24493.5i 1.03741i −0.854952 0.518707i \(-0.826414\pi\)
0.854952 0.518707i \(-0.173586\pi\)
\(824\) 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 0.490081 0.871677i \(-0.336967\pi\)
−0.871677 + 0.490081i \(0.836967\pi\)
\(830\) 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 \(-0.990864\pi\)
0.999588 0.0286991i \(-0.00913646\pi\)
\(840\) 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 −0.943863 0.330337i \(-0.892838\pi\)
0.330337 + 0.943863i \(0.392838\pi\)
\(854\) 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 \(-0.345031\pi\)
0.467843 + 0.883812i \(0.345031\pi\)
\(864\) 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 0.128139 0.991756i \(-0.459100\pi\)
−0.991756 + 0.128139i \(0.959100\pi\)
\(878\) 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 \(0.270834\pi\)
−0.659345 + 0.751841i \(0.729166\pi\)
\(888\) 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 \(-0.511660\pi\)
−0.0366219 + 0.999329i \(0.511660\pi\)
\(912\) 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 \(-0.763088\pi\)
0.735575 0.677443i \(-0.236912\pi\)
\(920\) 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 0.396012 0.918246i \(-0.370394\pi\)
−0.918246 + 0.396012i \(0.870394\pi\)
\(942\) 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 \(-0.738528\pi\)
0.681170 0.732126i \(-0.261472\pi\)
\(968\) 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 \(0.225098\pi\)
−0.760206 + 0.649682i \(0.774902\pi\)
\(984\) 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 \(-0.403254\pi\)
0.299280 + 0.954165i \(0.403254\pi\)
\(992\) 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 −0.964038 0.265764i \(-0.914376\pi\)
0.265764 + 0.964038i \(0.414376\pi\)
\(998\) 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 128.4.e.b.33.5 10
4.3 odd 2 128.4.e.a.33.1 10
8.3 odd 2 64.4.e.a.17.5 10
8.5 even 2 16.4.e.a.13.2 yes 10
16.3 odd 4 64.4.e.a.49.5 10
16.5 even 4 inner 128.4.e.b.97.5 10
16.11 odd 4 128.4.e.a.97.1 10
16.13 even 4 16.4.e.a.5.2 10
24.5 odd 2 144.4.k.a.109.4 10
24.11 even 2 576.4.k.a.145.1 10
32.3 odd 8 1024.4.b.k.513.1 10
32.5 even 8 1024.4.a.n.1.1 10
32.11 odd 8 1024.4.a.m.1.1 10
32.13 even 8 1024.4.b.j.513.1 10
32.19 odd 8 1024.4.b.k.513.10 10
32.21 even 8 1024.4.a.n.1.10 10
32.27 odd 8 1024.4.a.m.1.10 10
32.29 even 8 1024.4.b.j.513.10 10
48.29 odd 4 144.4.k.a.37.4 10
48.35 even 4 576.4.k.a.433.1 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.4.e.a.5.2 10 16.13 even 4
16.4.e.a.13.2 yes 10 8.5 even 2
64.4.e.a.17.5 10 8.3 odd 2
64.4.e.a.49.5 10 16.3 odd 4
128.4.e.a.33.1 10 4.3 odd 2
128.4.e.a.97.1 10 16.11 odd 4
128.4.e.b.33.5 10 1.1 even 1 trivial
128.4.e.b.97.5 10 16.5 even 4 inner
144.4.k.a.37.4 10 48.29 odd 4
144.4.k.a.109.4 10 24.5 odd 2
576.4.k.a.145.1 10 24.11 even 2
576.4.k.a.433.1 10 48.35 even 4
1024.4.a.m.1.1 10 32.11 odd 8
1024.4.a.m.1.10 10 32.27 odd 8
1024.4.a.n.1.1 10 32.5 even 8
1024.4.a.n.1.10 10 32.21 even 8
1024.4.b.j.513.1 10 32.13 even 8
1024.4.b.j.513.10 10 32.29 even 8
1024.4.b.k.513.1 10 32.3 odd 8
1024.4.b.k.513.10 10 32.19 odd 8