Properties

Label 624.4.d.b.287.10
Level $624$
Weight $4$
Character 624.287
Analytic conductor $36.817$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [624,4,Mod(287,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.287");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.d (of order \(2\), degree \(1\), 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: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 120x^{10} + 5196x^{8} + 96803x^{6} + 702900x^{4} + 976752x^{2} + 254016 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 287.10
Root \(3.92558i\) of defining polynomial
Character \(\chi\) \(=\) 624.287
Dual form 624.4.d.b.287.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(4.31028 + 2.90198i) q^{3} -9.74037i q^{5} +7.39870i q^{7} +(10.1570 + 25.0167i) q^{9} -40.2115 q^{11} -13.0000 q^{13} +(28.2664 - 41.9837i) q^{15} +69.5142i q^{17} -17.1044i q^{19} +(-21.4709 + 31.8904i) q^{21} -166.760 q^{23} +30.1252 q^{25} +(-28.8188 + 137.304i) q^{27} +250.179i q^{29} -234.386i q^{31} +(-173.323 - 116.693i) q^{33} +72.0660 q^{35} -373.210 q^{37} +(-56.0336 - 37.7258i) q^{39} +112.865i q^{41} +393.675i q^{43} +(243.672 - 98.9326i) q^{45} -227.157 q^{47} +288.259 q^{49} +(-201.729 + 299.625i) q^{51} +83.6563i q^{53} +391.675i q^{55} +(49.6367 - 73.7247i) q^{57} -120.477 q^{59} +706.102 q^{61} +(-185.091 + 75.1483i) q^{63} +126.625i q^{65} +647.317i q^{67} +(-718.783 - 483.936i) q^{69} -20.1069 q^{71} -388.286 q^{73} +(129.848 + 87.4230i) q^{75} -297.513i q^{77} +301.434i q^{79} +(-522.672 + 508.188i) q^{81} -1385.97 q^{83} +677.094 q^{85} +(-726.017 + 1078.34i) q^{87} +225.557i q^{89} -96.1830i q^{91} +(680.184 - 1010.27i) q^{93} -166.603 q^{95} -512.787 q^{97} +(-408.427 - 1005.96i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 54 q^{9} - 156 q^{13} - 54 q^{21} - 408 q^{25} - 360 q^{33} + 636 q^{37} - 810 q^{45} + 336 q^{49} - 1260 q^{57} + 960 q^{61} - 252 q^{69} + 3216 q^{73} - 2538 q^{81} + 2196 q^{85} - 1116 q^{93}+ \cdots + 4800 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
\(3\) 4.31028 + 2.90198i 0.829513 + 0.558487i
\(4\) 0 0
\(5\) 9.74037i 0.871205i −0.900139 0.435603i \(-0.856535\pi\)
0.900139 0.435603i \(-0.143465\pi\)
\(6\) 0 0
\(7\) 7.39870i 0.399492i 0.979848 + 0.199746i \(0.0640117\pi\)
−0.979848 + 0.199746i \(0.935988\pi\)
\(8\) 0 0
\(9\) 10.1570 + 25.0167i 0.376184 + 0.926545i
\(10\) 0 0
\(11\) −40.2115 −1.10220 −0.551101 0.834438i \(-0.685792\pi\)
−0.551101 + 0.834438i \(0.685792\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350
\(14\) 0 0
\(15\) 28.2664 41.9837i 0.486557 0.722676i
\(16\) 0 0
\(17\) 69.5142i 0.991745i 0.868395 + 0.495872i \(0.165152\pi\)
−0.868395 + 0.495872i \(0.834848\pi\)
\(18\) 0 0
\(19\) 17.1044i 0.206527i −0.994654 0.103264i \(-0.967071\pi\)
0.994654 0.103264i \(-0.0329285\pi\)
\(20\) 0 0
\(21\) −21.4709 + 31.8904i −0.223111 + 0.331384i
\(22\) 0 0
\(23\) −166.760 −1.51182 −0.755912 0.654674i \(-0.772806\pi\)
−0.755912 + 0.654674i \(0.772806\pi\)
\(24\) 0 0
\(25\) 30.1252 0.241002
\(26\) 0 0
\(27\) −28.8188 + 137.304i −0.205414 + 0.978675i
\(28\) 0 0
\(29\) 250.179i 1.60197i 0.598684 + 0.800985i \(0.295691\pi\)
−0.598684 + 0.800985i \(0.704309\pi\)
\(30\) 0 0
\(31\) 234.386i 1.35796i −0.734155 0.678982i \(-0.762421\pi\)
0.734155 0.678982i \(-0.237579\pi\)
\(32\) 0 0
\(33\) −173.323 116.693i −0.914292 0.615566i
\(34\) 0 0
\(35\) 72.0660 0.348039
\(36\) 0 0
\(37\) −373.210 −1.65825 −0.829126 0.559062i \(-0.811162\pi\)
−0.829126 + 0.559062i \(0.811162\pi\)
\(38\) 0 0
\(39\) −56.0336 37.7258i −0.230066 0.154896i
\(40\) 0 0
\(41\) 112.865i 0.429916i 0.976623 + 0.214958i \(0.0689615\pi\)
−0.976623 + 0.214958i \(0.931039\pi\)
\(42\) 0 0
\(43\) 393.675i 1.39616i 0.716021 + 0.698079i \(0.245962\pi\)
−0.716021 + 0.698079i \(0.754038\pi\)
\(44\) 0 0
\(45\) 243.672 98.9326i 0.807211 0.327733i
\(46\) 0 0
\(47\) −227.157 −0.704985 −0.352493 0.935815i \(-0.614666\pi\)
−0.352493 + 0.935815i \(0.614666\pi\)
\(48\) 0 0
\(49\) 288.259 0.840406
\(50\) 0 0
\(51\) −201.729 + 299.625i −0.553877 + 0.822665i
\(52\) 0 0
\(53\) 83.6563i 0.216813i 0.994107 + 0.108406i \(0.0345748\pi\)
−0.994107 + 0.108406i \(0.965425\pi\)
\(54\) 0 0
\(55\) 391.675i 0.960245i
\(56\) 0 0
\(57\) 49.6367 73.7247i 0.115343 0.171317i
\(58\) 0 0
\(59\) −120.477 −0.265844 −0.132922 0.991126i \(-0.542436\pi\)
−0.132922 + 0.991126i \(0.542436\pi\)
\(60\) 0 0
\(61\) 706.102 1.48208 0.741042 0.671458i \(-0.234332\pi\)
0.741042 + 0.671458i \(0.234332\pi\)
\(62\) 0 0
\(63\) −185.091 + 75.1483i −0.370147 + 0.150283i
\(64\) 0 0
\(65\) 126.625i 0.241629i
\(66\) 0 0
\(67\) 647.317i 1.18033i 0.807281 + 0.590167i \(0.200938\pi\)
−0.807281 + 0.590167i \(0.799062\pi\)
\(68\) 0 0
\(69\) −718.783 483.936i −1.25408 0.844334i
\(70\) 0 0
\(71\) −20.1069 −0.0336092 −0.0168046 0.999859i \(-0.505349\pi\)
−0.0168046 + 0.999859i \(0.505349\pi\)
\(72\) 0 0
\(73\) −388.286 −0.622540 −0.311270 0.950322i \(-0.600754\pi\)
−0.311270 + 0.950322i \(0.600754\pi\)
\(74\) 0 0
\(75\) 129.848 + 87.4230i 0.199914 + 0.134596i
\(76\) 0 0
\(77\) 297.513i 0.440321i
\(78\) 0 0
\(79\) 301.434i 0.429291i 0.976692 + 0.214646i \(0.0688597\pi\)
−0.976692 + 0.214646i \(0.931140\pi\)
\(80\) 0 0
\(81\) −522.672 + 508.188i −0.716971 + 0.697103i
\(82\) 0 0
\(83\) −1385.97 −1.83289 −0.916443 0.400166i \(-0.868952\pi\)
−0.916443 + 0.400166i \(0.868952\pi\)
\(84\) 0 0
\(85\) 677.094 0.864013
\(86\) 0 0
\(87\) −726.017 + 1078.34i −0.894680 + 1.32886i
\(88\) 0 0
\(89\) 225.557i 0.268640i 0.990938 + 0.134320i \(0.0428850\pi\)
−0.990938 + 0.134320i \(0.957115\pi\)
\(90\) 0 0
\(91\) 96.1830i 0.110799i
\(92\) 0 0
\(93\) 680.184 1010.27i 0.758406 1.12645i
\(94\) 0 0
\(95\) −166.603 −0.179928
\(96\) 0 0
\(97\) −512.787 −0.536759 −0.268379 0.963313i \(-0.586488\pi\)
−0.268379 + 0.963313i \(0.586488\pi\)
\(98\) 0 0
\(99\) −408.427 1005.96i −0.414631 1.02124i
\(100\) 0 0
\(101\) 397.173i 0.391289i −0.980675 0.195644i \(-0.937320\pi\)
0.980675 0.195644i \(-0.0626798\pi\)
\(102\) 0 0
\(103\) 1878.32i 1.79686i −0.439118 0.898429i \(-0.644709\pi\)
0.439118 0.898429i \(-0.355291\pi\)
\(104\) 0 0
\(105\) 310.624 + 209.135i 0.288703 + 0.194376i
\(106\) 0 0
\(107\) −310.026 −0.280106 −0.140053 0.990144i \(-0.544727\pi\)
−0.140053 + 0.990144i \(0.544727\pi\)
\(108\) 0 0
\(109\) 1416.14 1.24442 0.622211 0.782849i \(-0.286234\pi\)
0.622211 + 0.782849i \(0.286234\pi\)
\(110\) 0 0
\(111\) −1608.64 1083.05i −1.37554 0.926113i
\(112\) 0 0
\(113\) 2017.51i 1.67957i −0.542918 0.839786i \(-0.682680\pi\)
0.542918 0.839786i \(-0.317320\pi\)
\(114\) 0 0
\(115\) 1624.31i 1.31711i
\(116\) 0 0
\(117\) −132.041 325.217i −0.104335 0.256977i
\(118\) 0 0
\(119\) −514.314 −0.396194
\(120\) 0 0
\(121\) 285.966 0.214851
\(122\) 0 0
\(123\) −327.533 + 486.480i −0.240103 + 0.356621i
\(124\) 0 0
\(125\) 1510.98i 1.08117i
\(126\) 0 0
\(127\) 1694.52i 1.18397i 0.805948 + 0.591986i \(0.201656\pi\)
−0.805948 + 0.591986i \(0.798344\pi\)
\(128\) 0 0
\(129\) −1142.44 + 1696.85i −0.779737 + 1.15813i
\(130\) 0 0
\(131\) −1461.10 −0.974477 −0.487239 0.873269i \(-0.661996\pi\)
−0.487239 + 0.873269i \(0.661996\pi\)
\(132\) 0 0
\(133\) 126.550 0.0825060
\(134\) 0 0
\(135\) 1337.39 + 280.706i 0.852627 + 0.178958i
\(136\) 0 0
\(137\) 2397.25i 1.49497i 0.664278 + 0.747486i \(0.268739\pi\)
−0.664278 + 0.747486i \(0.731261\pi\)
\(138\) 0 0
\(139\) 1526.15i 0.931270i 0.884977 + 0.465635i \(0.154174\pi\)
−0.884977 + 0.465635i \(0.845826\pi\)
\(140\) 0 0
\(141\) −979.111 659.207i −0.584795 0.393725i
\(142\) 0 0
\(143\) 522.750 0.305696
\(144\) 0 0
\(145\) 2436.84 1.39565
\(146\) 0 0
\(147\) 1242.48 + 836.524i 0.697128 + 0.469356i
\(148\) 0 0
\(149\) 439.031i 0.241388i 0.992690 + 0.120694i \(0.0385120\pi\)
−0.992690 + 0.120694i \(0.961488\pi\)
\(150\) 0 0
\(151\) 520.091i 0.280294i 0.990131 + 0.140147i \(0.0447575\pi\)
−0.990131 + 0.140147i \(0.955243\pi\)
\(152\) 0 0
\(153\) −1739.02 + 706.053i −0.918896 + 0.373079i
\(154\) 0 0
\(155\) −2283.00 −1.18307
\(156\) 0 0
\(157\) −1732.36 −0.880620 −0.440310 0.897846i \(-0.645132\pi\)
−0.440310 + 0.897846i \(0.645132\pi\)
\(158\) 0 0
\(159\) −242.769 + 360.582i −0.121087 + 0.179849i
\(160\) 0 0
\(161\) 1233.81i 0.603961i
\(162\) 0 0
\(163\) 1359.90i 0.653471i −0.945116 0.326735i \(-0.894051\pi\)
0.945116 0.326735i \(-0.105949\pi\)
\(164\) 0 0
\(165\) −1136.63 + 1688.23i −0.536284 + 0.796535i
\(166\) 0 0
\(167\) −4.56410 −0.00211485 −0.00105743 0.999999i \(-0.500337\pi\)
−0.00105743 + 0.999999i \(0.500337\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) 427.896 173.729i 0.191357 0.0776923i
\(172\) 0 0
\(173\) 1543.85i 0.678478i 0.940700 + 0.339239i \(0.110169\pi\)
−0.940700 + 0.339239i \(0.889831\pi\)
\(174\) 0 0
\(175\) 222.887i 0.0962783i
\(176\) 0 0
\(177\) −519.291 349.624i −0.220521 0.148471i
\(178\) 0 0
\(179\) −130.180 −0.0543581 −0.0271790 0.999631i \(-0.508652\pi\)
−0.0271790 + 0.999631i \(0.508652\pi\)
\(180\) 0 0
\(181\) −702.455 −0.288470 −0.144235 0.989543i \(-0.546072\pi\)
−0.144235 + 0.989543i \(0.546072\pi\)
\(182\) 0 0
\(183\) 3043.50 + 2049.10i 1.22941 + 0.827725i
\(184\) 0 0
\(185\) 3635.20i 1.44468i
\(186\) 0 0
\(187\) 2795.27i 1.09310i
\(188\) 0 0
\(189\) −1015.87 213.221i −0.390973 0.0820613i
\(190\) 0 0
\(191\) 4299.41 1.62877 0.814384 0.580327i \(-0.197075\pi\)
0.814384 + 0.580327i \(0.197075\pi\)
\(192\) 0 0
\(193\) 3779.52 1.40962 0.704809 0.709398i \(-0.251033\pi\)
0.704809 + 0.709398i \(0.251033\pi\)
\(194\) 0 0
\(195\) −367.463 + 545.788i −0.134947 + 0.200434i
\(196\) 0 0
\(197\) 384.794i 0.139165i −0.997576 0.0695823i \(-0.977833\pi\)
0.997576 0.0695823i \(-0.0221666\pi\)
\(198\) 0 0
\(199\) 1016.18i 0.361984i 0.983485 + 0.180992i \(0.0579308\pi\)
−0.983485 + 0.180992i \(0.942069\pi\)
\(200\) 0 0
\(201\) −1878.50 + 2790.12i −0.659201 + 0.979102i
\(202\) 0 0
\(203\) −1851.00 −0.639975
\(204\) 0 0
\(205\) 1099.35 0.374545
\(206\) 0 0
\(207\) −1693.78 4171.80i −0.568724 1.40077i
\(208\) 0 0
\(209\) 687.794i 0.227635i
\(210\) 0 0
\(211\) 1416.76i 0.462246i 0.972925 + 0.231123i \(0.0742399\pi\)
−0.972925 + 0.231123i \(0.925760\pi\)
\(212\) 0 0
\(213\) −86.6665 58.3500i −0.0278793 0.0187703i
\(214\) 0 0
\(215\) 3834.53 1.21634
\(216\) 0 0
\(217\) 1734.15 0.542496
\(218\) 0 0
\(219\) −1673.62 1126.80i −0.516405 0.347681i
\(220\) 0 0
\(221\) 903.684i 0.275061i
\(222\) 0 0
\(223\) 5904.24i 1.77299i −0.462736 0.886496i \(-0.653132\pi\)
0.462736 0.886496i \(-0.346868\pi\)
\(224\) 0 0
\(225\) 305.981 + 753.634i 0.0906610 + 0.223299i
\(226\) 0 0
\(227\) 3060.83 0.894954 0.447477 0.894296i \(-0.352323\pi\)
0.447477 + 0.894296i \(0.352323\pi\)
\(228\) 0 0
\(229\) 3758.45 1.08457 0.542283 0.840196i \(-0.317560\pi\)
0.542283 + 0.840196i \(0.317560\pi\)
\(230\) 0 0
\(231\) 863.378 1282.36i 0.245914 0.365252i
\(232\) 0 0
\(233\) 3967.07i 1.11541i −0.830038 0.557706i \(-0.811682\pi\)
0.830038 0.557706i \(-0.188318\pi\)
\(234\) 0 0
\(235\) 2212.60i 0.614187i
\(236\) 0 0
\(237\) −874.758 + 1299.27i −0.239754 + 0.356103i
\(238\) 0 0
\(239\) 853.047 0.230874 0.115437 0.993315i \(-0.463173\pi\)
0.115437 + 0.993315i \(0.463173\pi\)
\(240\) 0 0
\(241\) −2985.05 −0.797858 −0.398929 0.916982i \(-0.630618\pi\)
−0.398929 + 0.916982i \(0.630618\pi\)
\(242\) 0 0
\(243\) −3727.61 + 673.644i −0.984060 + 0.177837i
\(244\) 0 0
\(245\) 2807.75i 0.732166i
\(246\) 0 0
\(247\) 222.357i 0.0572804i
\(248\) 0 0
\(249\) −5973.89 4022.05i −1.52040 1.02364i
\(250\) 0 0
\(251\) 4134.93 1.03982 0.519909 0.854222i \(-0.325966\pi\)
0.519909 + 0.854222i \(0.325966\pi\)
\(252\) 0 0
\(253\) 6705.69 1.66634
\(254\) 0 0
\(255\) 2918.46 + 1964.92i 0.716710 + 0.482540i
\(256\) 0 0
\(257\) 6118.58i 1.48508i −0.669800 0.742542i \(-0.733620\pi\)
0.669800 0.742542i \(-0.266380\pi\)
\(258\) 0 0
\(259\) 2761.27i 0.662459i
\(260\) 0 0
\(261\) −6258.67 + 2541.06i −1.48430 + 0.602636i
\(262\) 0 0
\(263\) 3717.92 0.871699 0.435850 0.900020i \(-0.356448\pi\)
0.435850 + 0.900020i \(0.356448\pi\)
\(264\) 0 0
\(265\) 814.843 0.188888
\(266\) 0 0
\(267\) −654.562 + 972.212i −0.150032 + 0.222841i
\(268\) 0 0
\(269\) 1363.32i 0.309007i 0.987992 + 0.154503i \(0.0493778\pi\)
−0.987992 + 0.154503i \(0.950622\pi\)
\(270\) 0 0
\(271\) 7954.61i 1.78306i −0.452966 0.891528i \(-0.649634\pi\)
0.452966 0.891528i \(-0.350366\pi\)
\(272\) 0 0
\(273\) 279.122 414.576i 0.0618799 0.0919094i
\(274\) 0 0
\(275\) −1211.38 −0.265633
\(276\) 0 0
\(277\) 998.868 0.216665 0.108332 0.994115i \(-0.465449\pi\)
0.108332 + 0.994115i \(0.465449\pi\)
\(278\) 0 0
\(279\) 5863.56 2380.65i 1.25822 0.510845i
\(280\) 0 0
\(281\) 6916.88i 1.46842i 0.678921 + 0.734211i \(0.262448\pi\)
−0.678921 + 0.734211i \(0.737552\pi\)
\(282\) 0 0
\(283\) 2364.19i 0.496596i −0.968684 0.248298i \(-0.920129\pi\)
0.968684 0.248298i \(-0.0798713\pi\)
\(284\) 0 0
\(285\) −718.106 483.480i −0.149252 0.100487i
\(286\) 0 0
\(287\) −835.054 −0.171748
\(288\) 0 0
\(289\) 80.7808 0.0164423
\(290\) 0 0
\(291\) −2210.25 1488.10i −0.445248 0.299773i
\(292\) 0 0
\(293\) 7233.06i 1.44218i 0.692839 + 0.721092i \(0.256360\pi\)
−0.692839 + 0.721092i \(0.743640\pi\)
\(294\) 0 0
\(295\) 1173.49i 0.231605i
\(296\) 0 0
\(297\) 1158.85 5521.22i 0.226408 1.07870i
\(298\) 0 0
\(299\) 2167.88 0.419304
\(300\) 0 0
\(301\) −2912.68 −0.557754
\(302\) 0 0
\(303\) 1152.59 1711.92i 0.218530 0.324579i
\(304\) 0 0
\(305\) 6877.70i 1.29120i
\(306\) 0 0
\(307\) 253.054i 0.0470441i −0.999723 0.0235220i \(-0.992512\pi\)
0.999723 0.0235220i \(-0.00748799\pi\)
\(308\) 0 0
\(309\) 5450.86 8096.08i 1.00352 1.49052i
\(310\) 0 0
\(311\) 374.616 0.0683039 0.0341519 0.999417i \(-0.489127\pi\)
0.0341519 + 0.999417i \(0.489127\pi\)
\(312\) 0 0
\(313\) 7578.97 1.36865 0.684327 0.729175i \(-0.260096\pi\)
0.684327 + 0.729175i \(0.260096\pi\)
\(314\) 0 0
\(315\) 731.972 + 1802.86i 0.130927 + 0.322474i
\(316\) 0 0
\(317\) 2992.77i 0.530255i −0.964213 0.265127i \(-0.914586\pi\)
0.964213 0.265127i \(-0.0854140\pi\)
\(318\) 0 0
\(319\) 10060.1i 1.76570i
\(320\) 0 0
\(321\) −1336.30 899.689i −0.232351 0.156435i
\(322\) 0 0
\(323\) 1189.00 0.204822
\(324\) 0 0
\(325\) −391.628 −0.0668419
\(326\) 0 0
\(327\) 6103.98 + 4109.63i 1.03226 + 0.694994i
\(328\) 0 0
\(329\) 1680.67i 0.281636i
\(330\) 0 0
\(331\) 8242.60i 1.36874i 0.729133 + 0.684372i \(0.239924\pi\)
−0.729133 + 0.684372i \(0.760076\pi\)
\(332\) 0 0
\(333\) −3790.68 9336.49i −0.623808 1.53645i
\(334\) 0 0
\(335\) 6305.11 1.02831
\(336\) 0 0
\(337\) 9334.33 1.50882 0.754412 0.656402i \(-0.227922\pi\)
0.754412 + 0.656402i \(0.227922\pi\)
\(338\) 0 0
\(339\) 5854.79 8696.04i 0.938020 1.39323i
\(340\) 0 0
\(341\) 9425.00i 1.49675i
\(342\) 0 0
\(343\) 4670.50i 0.735228i
\(344\) 0 0
\(345\) −4713.71 + 7001.21i −0.735588 + 1.09256i
\(346\) 0 0
\(347\) −3119.98 −0.482677 −0.241339 0.970441i \(-0.577586\pi\)
−0.241339 + 0.970441i \(0.577586\pi\)
\(348\) 0 0
\(349\) 7604.58 1.16637 0.583186 0.812339i \(-0.301806\pi\)
0.583186 + 0.812339i \(0.301806\pi\)
\(350\) 0 0
\(351\) 374.644 1784.96i 0.0569716 0.271436i
\(352\) 0 0
\(353\) 1079.00i 0.162689i 0.996686 + 0.0813445i \(0.0259214\pi\)
−0.996686 + 0.0813445i \(0.974079\pi\)
\(354\) 0 0
\(355\) 195.849i 0.0292805i
\(356\) 0 0
\(357\) −2216.84 1492.53i −0.328648 0.221269i
\(358\) 0 0
\(359\) −2495.63 −0.366892 −0.183446 0.983030i \(-0.558725\pi\)
−0.183446 + 0.983030i \(0.558725\pi\)
\(360\) 0 0
\(361\) 6566.44 0.957346
\(362\) 0 0
\(363\) 1232.59 + 829.870i 0.178222 + 0.119991i
\(364\) 0 0
\(365\) 3782.05i 0.542360i
\(366\) 0 0
\(367\) 4253.25i 0.604954i 0.953157 + 0.302477i \(0.0978135\pi\)
−0.953157 + 0.302477i \(0.902187\pi\)
\(368\) 0 0
\(369\) −2823.51 + 1146.37i −0.398337 + 0.161728i
\(370\) 0 0
\(371\) −618.947 −0.0866149
\(372\) 0 0
\(373\) −8121.75 −1.12742 −0.563711 0.825972i \(-0.690627\pi\)
−0.563711 + 0.825972i \(0.690627\pi\)
\(374\) 0 0
\(375\) 4384.83 6512.73i 0.603818 0.896842i
\(376\) 0 0
\(377\) 3252.33i 0.444307i
\(378\) 0 0
\(379\) 1385.36i 0.187761i 0.995583 + 0.0938804i \(0.0299271\pi\)
−0.995583 + 0.0938804i \(0.970073\pi\)
\(380\) 0 0
\(381\) −4917.47 + 7303.86i −0.661233 + 0.982120i
\(382\) 0 0
\(383\) −3745.59 −0.499714 −0.249857 0.968283i \(-0.580384\pi\)
−0.249857 + 0.968283i \(0.580384\pi\)
\(384\) 0 0
\(385\) −2897.88 −0.383610
\(386\) 0 0
\(387\) −9848.44 + 3998.54i −1.29360 + 0.525212i
\(388\) 0 0
\(389\) 4921.61i 0.641479i 0.947167 + 0.320740i \(0.103931\pi\)
−0.947167 + 0.320740i \(0.896069\pi\)
\(390\) 0 0
\(391\) 11592.2i 1.49934i
\(392\) 0 0
\(393\) −6297.73 4240.08i −0.808342 0.544233i
\(394\) 0 0
\(395\) 2936.08 0.374001
\(396\) 0 0
\(397\) −1356.50 −0.171489 −0.0857444 0.996317i \(-0.527327\pi\)
−0.0857444 + 0.996317i \(0.527327\pi\)
\(398\) 0 0
\(399\) 545.467 + 367.247i 0.0684398 + 0.0460786i
\(400\) 0 0
\(401\) 10471.3i 1.30402i −0.758209 0.652011i \(-0.773925\pi\)
0.758209 0.652011i \(-0.226075\pi\)
\(402\) 0 0
\(403\) 3047.01i 0.376632i
\(404\) 0 0
\(405\) 4949.94 + 5091.02i 0.607319 + 0.624629i
\(406\) 0 0
\(407\) 15007.3 1.82773
\(408\) 0 0
\(409\) −3404.75 −0.411624 −0.205812 0.978592i \(-0.565984\pi\)
−0.205812 + 0.978592i \(0.565984\pi\)
\(410\) 0 0
\(411\) −6956.79 + 10332.8i −0.834922 + 1.24010i
\(412\) 0 0
\(413\) 891.376i 0.106203i
\(414\) 0 0
\(415\) 13499.8i 1.59682i
\(416\) 0 0
\(417\) −4428.87 + 6578.14i −0.520103 + 0.772501i
\(418\) 0 0
\(419\) −11029.0 −1.28592 −0.642960 0.765899i \(-0.722294\pi\)
−0.642960 + 0.765899i \(0.722294\pi\)
\(420\) 0 0
\(421\) 307.919 0.0356462 0.0178231 0.999841i \(-0.494326\pi\)
0.0178231 + 0.999841i \(0.494326\pi\)
\(422\) 0 0
\(423\) −2307.23 5682.73i −0.265204 0.653201i
\(424\) 0 0
\(425\) 2094.13i 0.239012i
\(426\) 0 0
\(427\) 5224.24i 0.592081i
\(428\) 0 0
\(429\) 2253.20 + 1517.01i 0.253579 + 0.170727i
\(430\) 0 0
\(431\) 7719.32 0.862706 0.431353 0.902183i \(-0.358036\pi\)
0.431353 + 0.902183i \(0.358036\pi\)
\(432\) 0 0
\(433\) −998.047 −0.110769 −0.0553846 0.998465i \(-0.517638\pi\)
−0.0553846 + 0.998465i \(0.517638\pi\)
\(434\) 0 0
\(435\) 10503.5 + 7071.67i 1.15771 + 0.779450i
\(436\) 0 0
\(437\) 2852.34i 0.312233i
\(438\) 0 0
\(439\) 2305.00i 0.250596i −0.992119 0.125298i \(-0.960011\pi\)
0.992119 0.125298i \(-0.0399886\pi\)
\(440\) 0 0
\(441\) 2927.84 + 7211.30i 0.316147 + 0.778674i
\(442\) 0 0
\(443\) −10854.6 −1.16414 −0.582072 0.813137i \(-0.697758\pi\)
−0.582072 + 0.813137i \(0.697758\pi\)
\(444\) 0 0
\(445\) 2197.01 0.234041
\(446\) 0 0
\(447\) −1274.06 + 1892.34i −0.134812 + 0.200235i
\(448\) 0 0
\(449\) 15526.2i 1.63191i 0.578118 + 0.815953i \(0.303787\pi\)
−0.578118 + 0.815953i \(0.696213\pi\)
\(450\) 0 0
\(451\) 4538.48i 0.473855i
\(452\) 0 0
\(453\) −1509.29 + 2241.73i −0.156541 + 0.232507i
\(454\) 0 0
\(455\) −936.858 −0.0965288
\(456\) 0 0
\(457\) 10148.3 1.03877 0.519386 0.854540i \(-0.326161\pi\)
0.519386 + 0.854540i \(0.326161\pi\)
\(458\) 0 0
\(459\) −9544.60 2003.31i −0.970596 0.203718i
\(460\) 0 0
\(461\) 7974.58i 0.805668i 0.915273 + 0.402834i \(0.131975\pi\)
−0.915273 + 0.402834i \(0.868025\pi\)
\(462\) 0 0
\(463\) 9857.82i 0.989486i 0.869039 + 0.494743i \(0.164738\pi\)
−0.869039 + 0.494743i \(0.835262\pi\)
\(464\) 0 0
\(465\) −9840.37 6625.24i −0.981368 0.660727i
\(466\) 0 0
\(467\) −5020.39 −0.497464 −0.248732 0.968572i \(-0.580014\pi\)
−0.248732 + 0.968572i \(0.580014\pi\)
\(468\) 0 0
\(469\) −4789.30 −0.471534
\(470\) 0 0
\(471\) −7466.95 5027.28i −0.730486 0.491815i
\(472\) 0 0
\(473\) 15830.3i 1.53885i
\(474\) 0 0
\(475\) 515.274i 0.0497735i
\(476\) 0 0
\(477\) −2092.80 + 849.694i −0.200887 + 0.0815615i
\(478\) 0 0
\(479\) −13924.4 −1.32823 −0.664115 0.747631i \(-0.731191\pi\)
−0.664115 + 0.747631i \(0.731191\pi\)
\(480\) 0 0
\(481\) 4851.73 0.459916
\(482\) 0 0
\(483\) 3580.50 5318.06i 0.337305 0.500994i
\(484\) 0 0
\(485\) 4994.73i 0.467627i
\(486\) 0 0
\(487\) 17105.5i 1.59163i −0.605537 0.795817i \(-0.707042\pi\)
0.605537 0.795817i \(-0.292958\pi\)
\(488\) 0 0
\(489\) 3946.41 5861.55i 0.364955 0.542062i
\(490\) 0 0
\(491\) 3292.61 0.302634 0.151317 0.988485i \(-0.451649\pi\)
0.151317 + 0.988485i \(0.451649\pi\)
\(492\) 0 0
\(493\) −17391.0 −1.58875
\(494\) 0 0
\(495\) −9798.42 + 3978.23i −0.889710 + 0.361229i
\(496\) 0 0
\(497\) 148.765i 0.0134266i
\(498\) 0 0
\(499\) 7587.44i 0.680682i −0.940302 0.340341i \(-0.889457\pi\)
0.940302 0.340341i \(-0.110543\pi\)
\(500\) 0 0
\(501\) −19.6725 13.2449i −0.00175430 0.00118112i
\(502\) 0 0
\(503\) −15294.5 −1.35576 −0.677882 0.735170i \(-0.737102\pi\)
−0.677882 + 0.735170i \(0.737102\pi\)
\(504\) 0 0
\(505\) −3868.61 −0.340893
\(506\) 0 0
\(507\) 728.437 + 490.435i 0.0638087 + 0.0429606i
\(508\) 0 0
\(509\) 15799.9i 1.37587i −0.725771 0.687936i \(-0.758517\pi\)
0.725771 0.687936i \(-0.241483\pi\)
\(510\) 0 0
\(511\) 2872.81i 0.248700i
\(512\) 0 0
\(513\) 2348.51 + 492.928i 0.202123 + 0.0424236i
\(514\) 0 0
\(515\) −18295.5 −1.56543
\(516\) 0 0
\(517\) 9134.34 0.777037
\(518\) 0 0
\(519\) −4480.22 + 6654.41i −0.378921 + 0.562806i
\(520\) 0 0
\(521\) 13510.6i 1.13610i 0.822993 + 0.568052i \(0.192303\pi\)
−0.822993 + 0.568052i \(0.807697\pi\)
\(522\) 0 0
\(523\) 15580.7i 1.30267i 0.758790 + 0.651336i \(0.225791\pi\)
−0.758790 + 0.651336i \(0.774209\pi\)
\(524\) 0 0
\(525\) −646.816 + 960.706i −0.0537702 + 0.0798641i
\(526\) 0 0
\(527\) 16293.1 1.34675
\(528\) 0 0
\(529\) 15642.0 1.28561
\(530\) 0 0
\(531\) −1223.68 3013.95i −0.100006 0.246317i
\(532\) 0 0
\(533\) 1467.25i 0.119237i
\(534\) 0 0
\(535\) 3019.76i 0.244029i
\(536\) 0 0
\(537\) −561.111 377.780i −0.0450907 0.0303583i
\(538\) 0 0
\(539\) −11591.3 −0.926298
\(540\) 0 0
\(541\) 3006.57 0.238932 0.119466 0.992838i \(-0.461882\pi\)
0.119466 + 0.992838i \(0.461882\pi\)
\(542\) 0 0
\(543\) −3027.78 2038.51i −0.239290 0.161107i
\(544\) 0 0
\(545\) 13793.8i 1.08415i
\(546\) 0 0
\(547\) 10848.7i 0.848002i 0.905662 + 0.424001i \(0.139375\pi\)
−0.905662 + 0.424001i \(0.860625\pi\)
\(548\) 0 0
\(549\) 7171.86 + 17664.4i 0.557536 + 1.37322i
\(550\) 0 0
\(551\) 4279.17 0.330851
\(552\) 0 0
\(553\) −2230.22 −0.171499
\(554\) 0 0
\(555\) −10549.3 + 15668.7i −0.806834 + 1.19838i
\(556\) 0 0
\(557\) 23506.2i 1.78814i −0.447932 0.894068i \(-0.647839\pi\)
0.447932 0.894068i \(-0.352161\pi\)
\(558\) 0 0
\(559\) 5117.77i 0.387225i
\(560\) 0 0
\(561\) 8111.83 12048.4i 0.610485 0.906744i
\(562\) 0 0
\(563\) 5146.05 0.385222 0.192611 0.981275i \(-0.438304\pi\)
0.192611 + 0.981275i \(0.438304\pi\)
\(564\) 0 0
\(565\) −19651.3 −1.46325
\(566\) 0 0
\(567\) −3759.93 3867.09i −0.278487 0.286424i
\(568\) 0 0
\(569\) 18204.3i 1.34124i 0.741802 + 0.670619i \(0.233972\pi\)
−0.741802 + 0.670619i \(0.766028\pi\)
\(570\) 0 0
\(571\) 14558.7i 1.06701i 0.845797 + 0.533505i \(0.179125\pi\)
−0.845797 + 0.533505i \(0.820875\pi\)
\(572\) 0 0
\(573\) 18531.7 + 12476.8i 1.35108 + 0.909646i
\(574\) 0 0
\(575\) −5023.69 −0.364352
\(576\) 0 0
\(577\) 12638.2 0.911847 0.455924 0.890019i \(-0.349309\pi\)
0.455924 + 0.890019i \(0.349309\pi\)
\(578\) 0 0
\(579\) 16290.8 + 10968.1i 1.16930 + 0.787253i
\(580\) 0 0
\(581\) 10254.3i 0.732223i
\(582\) 0 0
\(583\) 3363.95i 0.238972i
\(584\) 0 0
\(585\) −3167.74 + 1286.12i −0.223880 + 0.0908969i
\(586\) 0 0
\(587\) −14489.0 −1.01878 −0.509390 0.860536i \(-0.670129\pi\)
−0.509390 + 0.860536i \(0.670129\pi\)
\(588\) 0 0
\(589\) −4009.03 −0.280457
\(590\) 0 0
\(591\) 1116.67 1658.57i 0.0777216 0.115439i
\(592\) 0 0
\(593\) 2580.48i 0.178697i 0.996000 + 0.0893487i \(0.0284785\pi\)
−0.996000 + 0.0893487i \(0.971521\pi\)
\(594\) 0 0
\(595\) 5009.61i 0.345166i
\(596\) 0 0
\(597\) −2948.93 + 4380.00i −0.202163 + 0.300270i
\(598\) 0 0
\(599\) 13625.6 0.929431 0.464715 0.885460i \(-0.346157\pi\)
0.464715 + 0.885460i \(0.346157\pi\)
\(600\) 0 0
\(601\) −16856.6 −1.14409 −0.572043 0.820223i \(-0.693849\pi\)
−0.572043 + 0.820223i \(0.693849\pi\)
\(602\) 0 0
\(603\) −16193.7 + 6574.78i −1.09363 + 0.444023i
\(604\) 0 0
\(605\) 2785.42i 0.187179i
\(606\) 0 0
\(607\) 7976.37i 0.533362i 0.963785 + 0.266681i \(0.0859270\pi\)
−0.963785 + 0.266681i \(0.914073\pi\)
\(608\) 0 0
\(609\) −7978.33 5371.58i −0.530867 0.357418i
\(610\) 0 0
\(611\) 2953.05 0.195528
\(612\) 0 0
\(613\) 881.451 0.0580774 0.0290387 0.999578i \(-0.490755\pi\)
0.0290387 + 0.999578i \(0.490755\pi\)
\(614\) 0 0
\(615\) 4738.49 + 3190.29i 0.310690 + 0.209179i
\(616\) 0 0
\(617\) 1846.48i 0.120481i 0.998184 + 0.0602404i \(0.0191867\pi\)
−0.998184 + 0.0602404i \(0.980813\pi\)
\(618\) 0 0
\(619\) 25535.3i 1.65808i 0.559189 + 0.829040i \(0.311113\pi\)
−0.559189 + 0.829040i \(0.688887\pi\)
\(620\) 0 0
\(621\) 4805.83 22896.9i 0.310550 1.47958i
\(622\) 0 0
\(623\) −1668.83 −0.107320
\(624\) 0 0
\(625\) −10951.8 −0.700916
\(626\) 0 0
\(627\) −1995.97 + 2964.58i −0.127131 + 0.188826i
\(628\) 0 0
\(629\) 25943.4i 1.64456i
\(630\) 0 0
\(631\) 17340.8i 1.09402i 0.837126 + 0.547009i \(0.184234\pi\)
−0.837126 + 0.547009i \(0.815766\pi\)
\(632\) 0 0
\(633\) −4111.42 + 6106.63i −0.258158 + 0.383439i
\(634\) 0 0
\(635\) 16505.3 1.03148
\(636\) 0 0
\(637\) −3747.37 −0.233087
\(638\) 0 0
\(639\) −204.226 503.010i −0.0126433 0.0311405i
\(640\) 0 0
\(641\) 1271.80i 0.0783668i 0.999232 + 0.0391834i \(0.0124757\pi\)
−0.999232 + 0.0391834i \(0.987524\pi\)
\(642\) 0 0
\(643\) 5662.47i 0.347288i −0.984808 0.173644i \(-0.944446\pi\)
0.984808 0.173644i \(-0.0555542\pi\)
\(644\) 0 0
\(645\) 16527.9 + 11127.8i 1.00897 + 0.679310i
\(646\) 0 0
\(647\) −22748.8 −1.38230 −0.691149 0.722712i \(-0.742895\pi\)
−0.691149 + 0.722712i \(0.742895\pi\)
\(648\) 0 0
\(649\) 4844.58 0.293014
\(650\) 0 0
\(651\) 7474.66 + 5032.47i 0.450008 + 0.302977i
\(652\) 0 0
\(653\) 727.610i 0.0436043i −0.999762 0.0218021i \(-0.993060\pi\)
0.999762 0.0218021i \(-0.00694038\pi\)
\(654\) 0 0
\(655\) 14231.6i 0.848969i
\(656\) 0 0
\(657\) −3943.81 9713.64i −0.234190 0.576811i
\(658\) 0 0
\(659\) −25989.0 −1.53625 −0.768126 0.640299i \(-0.778810\pi\)
−0.768126 + 0.640299i \(0.778810\pi\)
\(660\) 0 0
\(661\) −683.734 −0.0402333 −0.0201166 0.999798i \(-0.506404\pi\)
−0.0201166 + 0.999798i \(0.506404\pi\)
\(662\) 0 0
\(663\) 2622.48 3895.13i 0.153618 0.228166i
\(664\) 0 0
\(665\) 1232.65i 0.0718797i
\(666\) 0 0
\(667\) 41720.0i 2.42190i
\(668\) 0 0
\(669\) 17134.0 25448.9i 0.990194 1.47072i
\(670\) 0 0
\(671\) −28393.5 −1.63356
\(672\) 0 0
\(673\) −11692.5 −0.669705 −0.334853 0.942271i \(-0.608686\pi\)
−0.334853 + 0.942271i \(0.608686\pi\)
\(674\) 0 0
\(675\) −868.173 + 4136.32i −0.0495051 + 0.235863i
\(676\) 0 0
\(677\) 23194.7i 1.31676i −0.752686 0.658380i \(-0.771242\pi\)
0.752686 0.658380i \(-0.228758\pi\)
\(678\) 0 0
\(679\) 3793.95i 0.214431i
\(680\) 0 0
\(681\) 13193.0 + 8882.49i 0.742376 + 0.499820i
\(682\) 0 0
\(683\) 13096.4 0.733706 0.366853 0.930279i \(-0.380435\pi\)
0.366853 + 0.930279i \(0.380435\pi\)
\(684\) 0 0
\(685\) 23350.1 1.30243
\(686\) 0 0
\(687\) 16200.0 + 10907.0i 0.899661 + 0.605716i
\(688\) 0 0
\(689\) 1087.53i 0.0601330i
\(690\) 0 0
\(691\) 33382.5i 1.83782i −0.394472 0.918908i \(-0.629073\pi\)
0.394472 0.918908i \(-0.370927\pi\)
\(692\) 0 0
\(693\) 7442.79 3021.83i 0.407977 0.165642i
\(694\) 0 0
\(695\) 14865.3 0.811327
\(696\) 0 0
\(697\) −7845.72 −0.426367
\(698\) 0 0
\(699\) 11512.4 17099.1i 0.622944 0.925249i
\(700\) 0 0
\(701\) 17639.5i 0.950409i 0.879875 + 0.475204i \(0.157626\pi\)
−0.879875 + 0.475204i \(0.842374\pi\)
\(702\) 0 0
\(703\) 6383.53i 0.342474i
\(704\) 0 0
\(705\) −6420.92 + 9536.90i −0.343015 + 0.509476i
\(706\) 0 0
\(707\) 2938.56 0.156317
\(708\) 0 0
\(709\) −5863.12 −0.310570 −0.155285 0.987870i \(-0.549630\pi\)
−0.155285 + 0.987870i \(0.549630\pi\)
\(710\) 0 0
\(711\) −7540.90 + 3061.66i −0.397758 + 0.161493i
\(712\) 0 0
\(713\) 39086.2i 2.05300i
\(714\) 0 0
\(715\) 5091.78i 0.266324i
\(716\) 0 0
\(717\) 3676.87 + 2475.53i 0.191513 + 0.128940i
\(718\) 0 0
\(719\) −3072.01 −0.159342 −0.0796708 0.996821i \(-0.525387\pi\)
−0.0796708 + 0.996821i \(0.525387\pi\)
\(720\) 0 0
\(721\) 13897.1 0.717831
\(722\) 0 0
\(723\) −12866.4 8662.57i −0.661834 0.445594i
\(724\) 0 0
\(725\) 7536.71i 0.386078i
\(726\) 0 0
\(727\) 28176.7i 1.43743i −0.695303 0.718717i \(-0.744730\pi\)
0.695303 0.718717i \(-0.255270\pi\)
\(728\) 0 0
\(729\) −18022.0 7913.89i −0.915610 0.402067i
\(730\) 0 0
\(731\) −27366.0 −1.38463
\(732\) 0 0
\(733\) −28596.5 −1.44098 −0.720490 0.693466i \(-0.756083\pi\)
−0.720490 + 0.693466i \(0.756083\pi\)
\(734\) 0 0
\(735\) 8148.05 12102.2i 0.408905 0.607341i
\(736\) 0 0
\(737\) 26029.6i 1.30097i
\(738\) 0 0
\(739\) 35285.7i 1.75644i 0.478261 + 0.878218i \(0.341267\pi\)
−0.478261 + 0.878218i \(0.658733\pi\)
\(740\) 0 0
\(741\) −645.277 + 958.421i −0.0319904 + 0.0475148i
\(742\) 0 0
\(743\) 34165.6 1.68696 0.843482 0.537158i \(-0.180502\pi\)
0.843482 + 0.537158i \(0.180502\pi\)
\(744\) 0 0
\(745\) 4276.32 0.210298
\(746\) 0 0
\(747\) −14077.2 34672.3i −0.689502 1.69825i
\(748\) 0 0
\(749\) 2293.78i 0.111900i
\(750\) 0 0
\(751\) 32825.1i 1.59495i 0.603353 + 0.797474i \(0.293831\pi\)
−0.603353 + 0.797474i \(0.706169\pi\)
\(752\) 0 0
\(753\) 17822.7 + 11999.5i 0.862543 + 0.580725i
\(754\) 0 0
\(755\) 5065.87 0.244193
\(756\) 0 0
\(757\) −32213.7 −1.54667 −0.773334 0.633999i \(-0.781413\pi\)
−0.773334 + 0.633999i \(0.781413\pi\)
\(758\) 0 0
\(759\) 28903.4 + 19459.8i 1.38225 + 0.930627i
\(760\) 0 0
\(761\) 39231.3i 1.86877i 0.356264 + 0.934385i \(0.384051\pi\)
−0.356264 + 0.934385i \(0.615949\pi\)
\(762\) 0 0
\(763\) 10477.6i 0.497137i
\(764\) 0 0
\(765\) 6877.22 + 16938.7i 0.325028 + 0.800547i
\(766\) 0 0
\(767\) 1566.21 0.0737320
\(768\) 0 0
\(769\) −20637.3 −0.967752 −0.483876 0.875137i \(-0.660771\pi\)
−0.483876 + 0.875137i \(0.660771\pi\)
\(770\) 0 0
\(771\) 17756.0 26372.8i 0.829400 1.23190i
\(772\) 0 0
\(773\) 14291.6i 0.664986i 0.943106 + 0.332493i \(0.107890\pi\)
−0.943106 + 0.332493i \(0.892110\pi\)
\(774\) 0 0
\(775\) 7060.92i 0.327272i
\(776\) 0 0
\(777\) 8013.15 11901.8i 0.369975 0.549518i
\(778\) 0 0
\(779\) 1930.49 0.0887894
\(780\) 0 0
\(781\) 808.531 0.0370442
\(782\) 0 0
\(783\) −34350.7 7209.87i −1.56781 0.329067i
\(784\) 0 0
\(785\) 16873.8i 0.767201i
\(786\) 0 0
\(787\) 31135.5i 1.41024i −0.709088 0.705120i \(-0.750893\pi\)
0.709088 0.705120i \(-0.249107\pi\)
\(788\) 0 0
\(789\) 16025.3 + 10789.4i 0.723086 + 0.486833i
\(790\) 0 0
\(791\) 14927.0 0.670976
\(792\) 0 0
\(793\) −9179.33 −0.411056
\(794\) 0 0
\(795\) 3512.20 + 2364.66i 0.156685 + 0.105492i
\(796\) 0 0
\(797\) 17943.6i 0.797485i 0.917063 + 0.398742i \(0.130553\pi\)
−0.917063 + 0.398742i \(0.869447\pi\)
\(798\) 0 0
\(799\) 15790.7i 0.699166i
\(800\) 0 0
\(801\) −5642.69 + 2290.97i −0.248907 + 0.101058i
\(802\) 0 0
\(803\) 15613.6 0.686165
\(804\) 0 0
\(805\) −12017.8 −0.526174
\(806\) 0 0
\(807\) −3956.32 + 5876.27i −0.172576 + 0.256325i
\(808\) 0 0
\(809\) 10039.8i 0.436317i −0.975913 0.218159i \(-0.929995\pi\)
0.975913 0.218159i \(-0.0700050\pi\)
\(810\) 0 0
\(811\) 31349.9i 1.35739i −0.734420 0.678695i \(-0.762546\pi\)
0.734420 0.678695i \(-0.237454\pi\)
\(812\) 0 0
\(813\) 23084.2 34286.6i 0.995814 1.47907i
\(814\) 0 0
\(815\) −13245.9 −0.569307
\(816\) 0 0
\(817\) 6733.57 0.288345
\(818\) 0 0
\(819\) 2406.18 976.928i 0.102660 0.0416809i
\(820\) 0 0
\(821\) 36012.2i 1.53086i 0.643521 + 0.765429i \(0.277473\pi\)
−0.643521 + 0.765429i \(0.722527\pi\)
\(822\) 0 0
\(823\) 5750.21i 0.243548i −0.992558 0.121774i \(-0.961142\pi\)
0.992558 0.121774i \(-0.0388583\pi\)
\(824\) 0 0
\(825\) −5221.39 3515.41i −0.220346 0.148353i
\(826\) 0 0
\(827\) 2668.08 0.112186 0.0560932 0.998426i \(-0.482136\pi\)
0.0560932 + 0.998426i \(0.482136\pi\)
\(828\) 0 0
\(829\) −20803.0 −0.871553 −0.435777 0.900055i \(-0.643526\pi\)
−0.435777 + 0.900055i \(0.643526\pi\)
\(830\) 0 0
\(831\) 4305.40 + 2898.70i 0.179726 + 0.121005i
\(832\) 0 0
\(833\) 20038.1i 0.833468i
\(834\) 0 0
\(835\) 44.4560i 0.00184247i
\(836\) 0 0
\(837\) 32182.2 + 6754.71i 1.32901 + 0.278945i
\(838\) 0 0
\(839\) 44034.2 1.81196 0.905978 0.423325i \(-0.139137\pi\)
0.905978 + 0.423325i \(0.139137\pi\)
\(840\) 0 0
\(841\) −38200.7 −1.56631
\(842\) 0 0
\(843\) −20072.7 + 29813.7i −0.820095 + 1.21808i
\(844\) 0 0
\(845\) 1646.12i 0.0670158i
\(846\) 0 0
\(847\) 2115.78i 0.0858312i
\(848\) 0 0
\(849\) 6860.85 10190.3i 0.277343 0.411933i
\(850\) 0 0
\(851\) 62236.6 2.50698
\(852\) 0 0
\(853\) −22838.0 −0.916715 −0.458357 0.888768i \(-0.651562\pi\)
−0.458357 + 0.888768i \(0.651562\pi\)
\(854\) 0 0
\(855\) −1692.18 4167.86i −0.0676859 0.166711i
\(856\) 0 0
\(857\) 33000.9i 1.31539i −0.753284 0.657695i \(-0.771532\pi\)
0.753284 0.657695i \(-0.228468\pi\)
\(858\) 0 0
\(859\) 9649.22i 0.383268i −0.981466 0.191634i \(-0.938621\pi\)
0.981466 0.191634i \(-0.0613787\pi\)
\(860\) 0 0
\(861\) −3599.32 2423.32i −0.142467 0.0959191i
\(862\) 0 0
\(863\) −2249.42 −0.0887266 −0.0443633 0.999015i \(-0.514126\pi\)
−0.0443633 + 0.999015i \(0.514126\pi\)
\(864\) 0 0
\(865\) 15037.7 0.591093
\(866\) 0 0
\(867\) 348.188 + 234.425i 0.0136391 + 0.00918279i
\(868\) 0 0
\(869\) 12121.1i 0.473166i
\(870\) 0 0
\(871\) 8415.12i 0.327366i
\(872\) 0 0
\(873\) −5208.36 12828.2i −0.201920 0.497331i
\(874\) 0 0
\(875\) 11179.3 0.431918
\(876\) 0 0
\(877\) −562.158 −0.0216451 −0.0108225 0.999941i \(-0.503445\pi\)
−0.0108225 + 0.999941i \(0.503445\pi\)
\(878\) 0 0
\(879\) −20990.2 + 31176.5i −0.805442 + 1.19631i
\(880\) 0 0
\(881\) 14155.3i 0.541320i 0.962675 + 0.270660i \(0.0872419\pi\)
−0.962675 + 0.270660i \(0.912758\pi\)
\(882\) 0 0
\(883\) 51391.0i 1.95860i 0.202413 + 0.979300i \(0.435122\pi\)
−0.202413 + 0.979300i \(0.564878\pi\)
\(884\) 0 0
\(885\) −3405.46 + 5058.08i −0.129348 + 0.192119i
\(886\) 0 0
\(887\) −24844.4 −0.940467 −0.470233 0.882542i \(-0.655830\pi\)
−0.470233 + 0.882542i \(0.655830\pi\)
\(888\) 0 0
\(889\) −12537.2 −0.472987
\(890\) 0 0
\(891\) 21017.4 20435.0i 0.790248 0.768349i
\(892\) 0 0
\(893\) 3885.39i 0.145599i
\(894\) 0 0
\(895\) 1268.00i 0.0473570i
\(896\) 0 0
\(897\) 9344.18 + 6291.17i 0.347818 + 0.234176i
\(898\) 0 0
\(899\) 58638.5 2.17542
\(900\) 0 0
\(901\) −5815.30 −0.215023
\(902\) 0 0
\(903\) −12554.4 8452.55i −0.462664 0.311499i
\(904\) 0 0
\(905\) 6842.17i 0.251317i
\(906\) 0 0
\(907\) 7191.92i 0.263290i −0.991297 0.131645i \(-0.957974\pi\)
0.991297 0.131645i \(-0.0420258\pi\)
\(908\) 0 0
\(909\) 9935.96 4034.07i 0.362547 0.147197i
\(910\) 0 0
\(911\) 17622.8 0.640910 0.320455 0.947264i \(-0.396164\pi\)
0.320455 + 0.947264i \(0.396164\pi\)
\(912\) 0 0
\(913\) 55731.8 2.02021
\(914\) 0 0
\(915\) 19959.0 29644.8i 0.721118 1.07107i
\(916\) 0 0
\(917\) 10810.2i 0.389296i
\(918\) 0 0
\(919\) 21839.1i 0.783902i 0.919986 + 0.391951i \(0.128200\pi\)
−0.919986 + 0.391951i \(0.871800\pi\)
\(920\) 0 0
\(921\) 734.358 1090.73i 0.0262735 0.0390237i
\(922\) 0 0
\(923\) 261.390 0.00932152
\(924\) 0 0
\(925\) −11243.0 −0.399642
\(926\) 0 0
\(927\) 46989.4 19078.0i 1.66487 0.675949i
\(928\) 0 0
\(929\) 8108.45i 0.286361i 0.989697 + 0.143181i \(0.0457330\pi\)
−0.989697 + 0.143181i \(0.954267\pi\)
\(930\) 0 0
\(931\) 4930.50i 0.173567i
\(932\) 0 0
\(933\) 1614.70 + 1087.13i 0.0566589 + 0.0381468i
\(934\) 0 0
\(935\) −27227.0 −0.952318
\(936\) 0 0
\(937\) 25761.6 0.898180 0.449090 0.893487i \(-0.351748\pi\)
0.449090 + 0.893487i \(0.351748\pi\)
\(938\) 0 0
\(939\) 32667.4 + 21994.0i 1.13532 + 0.764376i
\(940\) 0 0
\(941\) 8189.74i 0.283717i 0.989887 + 0.141859i \(0.0453078\pi\)
−0.989887 + 0.141859i \(0.954692\pi\)
\(942\) 0 0
\(943\) 18821.4i 0.649957i
\(944\) 0 0
\(945\) −2076.86 + 9894.98i −0.0714922 + 0.340618i
\(946\) 0 0
\(947\) 55018.7 1.88793 0.943964 0.330050i \(-0.107065\pi\)
0.943964 + 0.330050i \(0.107065\pi\)
\(948\) 0 0
\(949\) 5047.72 0.172662
\(950\) 0 0
\(951\) 8684.98 12899.7i 0.296141 0.439853i
\(952\) 0 0
\(953\) 36617.4i 1.24465i −0.782758 0.622327i \(-0.786188\pi\)
0.782758 0.622327i \(-0.213812\pi\)
\(954\) 0 0
\(955\) 41877.9i 1.41899i
\(956\) 0 0
\(957\) 29194.2 43361.8i 0.986119 1.46467i
\(958\) 0 0
\(959\) −17736.5 −0.597229
\(960\) 0 0
\(961\) −25145.6 −0.844068
\(962\) 0 0
\(963\) −3148.92 7755.82i −0.105371 0.259530i
\(964\) 0 0
\(965\) 36814.0i 1.22807i
\(966\) 0 0
\(967\) 7464.23i 0.248225i −0.992268 0.124112i \(-0.960392\pi\)
0.992268 0.124112i \(-0.0396083\pi\)
\(968\) 0 0
\(969\) 5124.91 + 3450.45i 0.169903 + 0.114391i
\(970\) 0 0
\(971\) −36787.0 −1.21581 −0.607906 0.794009i \(-0.707990\pi\)
−0.607906 + 0.794009i \(0.707990\pi\)
\(972\) 0 0
\(973\) −11291.5 −0.372035
\(974\) 0 0
\(975\) −1688.02 1136.50i −0.0554462 0.0373303i
\(976\) 0 0
\(977\) 27402.6i 0.897326i −0.893701 0.448663i \(-0.851900\pi\)
0.893701 0.448663i \(-0.148100\pi\)
\(978\) 0 0
\(979\) 9069.98i 0.296096i
\(980\) 0 0
\(981\) 14383.7 + 35427.3i 0.468132 + 1.15301i
\(982\) 0 0
\(983\) −53135.8 −1.72408 −0.862040 0.506841i \(-0.830813\pi\)
−0.862040 + 0.506841i \(0.830813\pi\)
\(984\) 0 0
\(985\) −3748.03 −0.121241
\(986\) 0 0
\(987\) 4877.27 7244.15i 0.157290 0.233621i
\(988\) 0 0
\(989\) 65649.3i 2.11074i
\(990\) 0 0
\(991\) 10055.2i 0.322315i −0.986929 0.161158i \(-0.948477\pi\)
0.986929 0.161158i \(-0.0515228\pi\)
\(992\) 0 0
\(993\) −23919.9 + 35527.9i −0.764426 + 1.13539i
\(994\) 0 0
\(995\) 9897.92 0.315362
\(996\) 0 0
\(997\) −39092.2 −1.24179 −0.620894 0.783895i \(-0.713230\pi\)
−0.620894 + 0.783895i \(0.713230\pi\)
\(998\) 0 0
\(999\) 10755.5 51243.3i 0.340628 1.62289i
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 624.4.d.b.287.10 yes 12
3.2 odd 2 inner 624.4.d.b.287.4 yes 12
4.3 odd 2 inner 624.4.d.b.287.3 12
12.11 even 2 inner 624.4.d.b.287.9 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
624.4.d.b.287.3 12 4.3 odd 2 inner
624.4.d.b.287.4 yes 12 3.2 odd 2 inner
624.4.d.b.287.9 yes 12 12.11 even 2 inner
624.4.d.b.287.10 yes 12 1.1 even 1 trivial