Properties

Label 960.3.i.b.929.37
Level $960$
Weight $3$
Character 960.929
Analytic conductor $26.158$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [960,3,Mod(929,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.929");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 960.i (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: \(26.1581053786\)
Analytic rank: \(0\)
Dimension: \(64\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 929.37
Character \(\chi\) \(=\) 960.929
Dual form 960.3.i.b.929.34

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.619753 + 2.93529i) q^{3} +(-4.48354 - 2.21312i) q^{5} +4.63217i q^{7} +(-8.23181 + 3.63830i) q^{9} +O(q^{10})\) \(q+(0.619753 + 2.93529i) q^{3} +(-4.48354 - 2.21312i) q^{5} +4.63217i q^{7} +(-8.23181 + 3.63830i) q^{9} +15.0818 q^{11} +3.35922 q^{13} +(3.71746 - 14.5320i) q^{15} +11.1429 q^{17} +22.7628i q^{19} +(-13.5967 + 2.87080i) q^{21} +36.1566 q^{23} +(15.2042 + 19.8452i) q^{25} +(-15.7811 - 21.9079i) q^{27} -21.5705 q^{29} -28.3890 q^{31} +(9.34696 + 44.2693i) q^{33} +(10.2516 - 20.7685i) q^{35} -69.5592 q^{37} +(2.08189 + 9.86027i) q^{39} +62.5531i q^{41} -55.2064 q^{43} +(44.9596 + 1.90555i) q^{45} -36.5257 q^{47} +27.5430 q^{49} +(6.90582 + 32.7075i) q^{51} -29.2870i q^{53} +(-67.6196 - 33.3778i) q^{55} +(-66.8152 + 14.1073i) q^{57} -8.07476 q^{59} -10.5504i q^{61} +(-16.8532 - 38.1312i) q^{63} +(-15.0612 - 7.43436i) q^{65} -91.8695 q^{67} +(22.4082 + 106.130i) q^{69} +79.2826i q^{71} +59.0857i q^{73} +(-48.8286 + 56.9278i) q^{75} +69.8613i q^{77} +86.3194 q^{79} +(54.5255 - 59.8997i) q^{81} -10.0197i q^{83} +(-49.9594 - 24.6605i) q^{85} +(-13.3684 - 63.3157i) q^{87} -17.8051i q^{89} +15.5605i q^{91} +(-17.5941 - 83.3297i) q^{93} +(50.3767 - 102.058i) q^{95} +139.268i q^{97} +(-124.150 + 54.8720i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 64 q - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 64 q - 32 q^{9} - 32 q^{25} - 320 q^{49} + 448 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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
\(3\) 0.619753 + 2.93529i 0.206584 + 0.978429i
\(4\) 0 0
\(5\) −4.48354 2.21312i −0.896707 0.442624i
\(6\) 0 0
\(7\) 4.63217i 0.661739i 0.943677 + 0.330869i \(0.107342\pi\)
−0.943677 + 0.330869i \(0.892658\pi\)
\(8\) 0 0
\(9\) −8.23181 + 3.63830i −0.914646 + 0.404256i
\(10\) 0 0
\(11\) 15.0818 1.37107 0.685534 0.728040i \(-0.259569\pi\)
0.685534 + 0.728040i \(0.259569\pi\)
\(12\) 0 0
\(13\) 3.35922 0.258402 0.129201 0.991618i \(-0.458759\pi\)
0.129201 + 0.991618i \(0.458759\pi\)
\(14\) 0 0
\(15\) 3.71746 14.5320i 0.247831 0.968803i
\(16\) 0 0
\(17\) 11.1429 0.655463 0.327731 0.944771i \(-0.393716\pi\)
0.327731 + 0.944771i \(0.393716\pi\)
\(18\) 0 0
\(19\) 22.7628i 1.19804i 0.800734 + 0.599020i \(0.204443\pi\)
−0.800734 + 0.599020i \(0.795557\pi\)
\(20\) 0 0
\(21\) −13.5967 + 2.87080i −0.647464 + 0.136705i
\(22\) 0 0
\(23\) 36.1566 1.57203 0.786014 0.618209i \(-0.212142\pi\)
0.786014 + 0.618209i \(0.212142\pi\)
\(24\) 0 0
\(25\) 15.2042 + 19.8452i 0.608167 + 0.793809i
\(26\) 0 0
\(27\) −15.7811 21.9079i −0.584487 0.811403i
\(28\) 0 0
\(29\) −21.5705 −0.743811 −0.371906 0.928271i \(-0.621295\pi\)
−0.371906 + 0.928271i \(0.621295\pi\)
\(30\) 0 0
\(31\) −28.3890 −0.915773 −0.457886 0.889011i \(-0.651393\pi\)
−0.457886 + 0.889011i \(0.651393\pi\)
\(32\) 0 0
\(33\) 9.34696 + 44.2693i 0.283241 + 1.34149i
\(34\) 0 0
\(35\) 10.2516 20.7685i 0.292902 0.593386i
\(36\) 0 0
\(37\) −69.5592 −1.87998 −0.939989 0.341205i \(-0.889165\pi\)
−0.939989 + 0.341205i \(0.889165\pi\)
\(38\) 0 0
\(39\) 2.08189 + 9.86027i 0.0533817 + 0.252828i
\(40\) 0 0
\(41\) 62.5531i 1.52568i 0.646585 + 0.762842i \(0.276197\pi\)
−0.646585 + 0.762842i \(0.723803\pi\)
\(42\) 0 0
\(43\) −55.2064 −1.28387 −0.641935 0.766759i \(-0.721868\pi\)
−0.641935 + 0.766759i \(0.721868\pi\)
\(44\) 0 0
\(45\) 44.9596 + 1.90555i 0.999103 + 0.0423455i
\(46\) 0 0
\(47\) −36.5257 −0.777142 −0.388571 0.921419i \(-0.627031\pi\)
−0.388571 + 0.921419i \(0.627031\pi\)
\(48\) 0 0
\(49\) 27.5430 0.562102
\(50\) 0 0
\(51\) 6.90582 + 32.7075i 0.135408 + 0.641323i
\(52\) 0 0
\(53\) 29.2870i 0.552585i −0.961074 0.276292i \(-0.910894\pi\)
0.961074 0.276292i \(-0.0891058\pi\)
\(54\) 0 0
\(55\) −67.6196 33.3778i −1.22945 0.606868i
\(56\) 0 0
\(57\) −66.8152 + 14.1073i −1.17220 + 0.247496i
\(58\) 0 0
\(59\) −8.07476 −0.136860 −0.0684302 0.997656i \(-0.521799\pi\)
−0.0684302 + 0.997656i \(0.521799\pi\)
\(60\) 0 0
\(61\) 10.5504i 0.172957i −0.996254 0.0864783i \(-0.972439\pi\)
0.996254 0.0864783i \(-0.0275613\pi\)
\(62\) 0 0
\(63\) −16.8532 38.1312i −0.267512 0.605257i
\(64\) 0 0
\(65\) −15.0612 7.43436i −0.231711 0.114375i
\(66\) 0 0
\(67\) −91.8695 −1.37119 −0.685593 0.727985i \(-0.740457\pi\)
−0.685593 + 0.727985i \(0.740457\pi\)
\(68\) 0 0
\(69\) 22.4082 + 106.130i 0.324756 + 1.53812i
\(70\) 0 0
\(71\) 79.2826i 1.11666i 0.829620 + 0.558328i \(0.188557\pi\)
−0.829620 + 0.558328i \(0.811443\pi\)
\(72\) 0 0
\(73\) 59.0857i 0.809394i 0.914451 + 0.404697i \(0.132623\pi\)
−0.914451 + 0.404697i \(0.867377\pi\)
\(74\) 0 0
\(75\) −48.8286 + 56.9278i −0.651048 + 0.759037i
\(76\) 0 0
\(77\) 69.8613i 0.907289i
\(78\) 0 0
\(79\) 86.3194 1.09265 0.546325 0.837573i \(-0.316026\pi\)
0.546325 + 0.837573i \(0.316026\pi\)
\(80\) 0 0
\(81\) 54.5255 59.8997i 0.673154 0.739502i
\(82\) 0 0
\(83\) 10.0197i 0.120720i −0.998177 0.0603599i \(-0.980775\pi\)
0.998177 0.0603599i \(-0.0192248\pi\)
\(84\) 0 0
\(85\) −49.9594 24.6605i −0.587758 0.290124i
\(86\) 0 0
\(87\) −13.3684 63.3157i −0.153660 0.727766i
\(88\) 0 0
\(89\) 17.8051i 0.200057i −0.994985 0.100029i \(-0.968107\pi\)
0.994985 0.100029i \(-0.0318935\pi\)
\(90\) 0 0
\(91\) 15.5605i 0.170994i
\(92\) 0 0
\(93\) −17.5941 83.3297i −0.189184 0.896019i
\(94\) 0 0
\(95\) 50.3767 102.058i 0.530282 1.07429i
\(96\) 0 0
\(97\) 139.268i 1.43575i 0.696170 + 0.717877i \(0.254886\pi\)
−0.696170 + 0.717877i \(0.745114\pi\)
\(98\) 0 0
\(99\) −124.150 + 54.8720i −1.25404 + 0.554263i
\(100\) 0 0
\(101\) 46.6948 0.462325 0.231162 0.972915i \(-0.425747\pi\)
0.231162 + 0.972915i \(0.425747\pi\)
\(102\) 0 0
\(103\) 11.6291i 0.112904i −0.998405 0.0564520i \(-0.982021\pi\)
0.998405 0.0564520i \(-0.0179788\pi\)
\(104\) 0 0
\(105\) 67.3149 + 17.2199i 0.641095 + 0.163999i
\(106\) 0 0
\(107\) 95.7996i 0.895323i −0.894203 0.447662i \(-0.852257\pi\)
0.894203 0.447662i \(-0.147743\pi\)
\(108\) 0 0
\(109\) 124.614i 1.14324i 0.820517 + 0.571622i \(0.193686\pi\)
−0.820517 + 0.571622i \(0.806314\pi\)
\(110\) 0 0
\(111\) −43.1095 204.176i −0.388374 1.83942i
\(112\) 0 0
\(113\) 9.29116 0.0822227 0.0411113 0.999155i \(-0.486910\pi\)
0.0411113 + 0.999155i \(0.486910\pi\)
\(114\) 0 0
\(115\) −162.110 80.0190i −1.40965 0.695817i
\(116\) 0 0
\(117\) −27.6525 + 12.2219i −0.236346 + 0.104460i
\(118\) 0 0
\(119\) 51.6156i 0.433745i
\(120\) 0 0
\(121\) 106.459 0.879830
\(122\) 0 0
\(123\) −183.611 + 38.7674i −1.49277 + 0.315182i
\(124\) 0 0
\(125\) −24.2486 122.625i −0.193989 0.981004i
\(126\) 0 0
\(127\) 167.887i 1.32194i −0.750412 0.660971i \(-0.770145\pi\)
0.750412 0.660971i \(-0.229855\pi\)
\(128\) 0 0
\(129\) −34.2143 162.047i −0.265227 1.25618i
\(130\) 0 0
\(131\) −6.27609 −0.0479091 −0.0239546 0.999713i \(-0.507626\pi\)
−0.0239546 + 0.999713i \(0.507626\pi\)
\(132\) 0 0
\(133\) −105.441 −0.792789
\(134\) 0 0
\(135\) 22.2705 + 133.150i 0.164967 + 0.986299i
\(136\) 0 0
\(137\) −100.937 −0.736764 −0.368382 0.929675i \(-0.620088\pi\)
−0.368382 + 0.929675i \(0.620088\pi\)
\(138\) 0 0
\(139\) 182.361i 1.31195i 0.754782 + 0.655976i \(0.227743\pi\)
−0.754782 + 0.655976i \(0.772257\pi\)
\(140\) 0 0
\(141\) −22.6369 107.213i −0.160545 0.760378i
\(142\) 0 0
\(143\) 50.6629 0.354286
\(144\) 0 0
\(145\) 96.7122 + 47.7382i 0.666981 + 0.329229i
\(146\) 0 0
\(147\) 17.0698 + 80.8466i 0.116121 + 0.549977i
\(148\) 0 0
\(149\) −207.094 −1.38989 −0.694946 0.719062i \(-0.744572\pi\)
−0.694946 + 0.719062i \(0.744572\pi\)
\(150\) 0 0
\(151\) −51.4246 −0.340560 −0.170280 0.985396i \(-0.554467\pi\)
−0.170280 + 0.985396i \(0.554467\pi\)
\(152\) 0 0
\(153\) −91.7260 + 40.5411i −0.599516 + 0.264975i
\(154\) 0 0
\(155\) 127.283 + 62.8282i 0.821180 + 0.405343i
\(156\) 0 0
\(157\) 143.715 0.915385 0.457692 0.889111i \(-0.348676\pi\)
0.457692 + 0.889111i \(0.348676\pi\)
\(158\) 0 0
\(159\) 85.9657 18.1507i 0.540665 0.114155i
\(160\) 0 0
\(161\) 167.484i 1.04027i
\(162\) 0 0
\(163\) 120.403 0.738669 0.369335 0.929296i \(-0.379586\pi\)
0.369335 + 0.929296i \(0.379586\pi\)
\(164\) 0 0
\(165\) 56.0659 219.169i 0.339793 1.32830i
\(166\) 0 0
\(167\) −117.491 −0.703540 −0.351770 0.936086i \(-0.614420\pi\)
−0.351770 + 0.936086i \(0.614420\pi\)
\(168\) 0 0
\(169\) −157.716 −0.933229
\(170\) 0 0
\(171\) −82.8178 187.379i −0.484315 1.09578i
\(172\) 0 0
\(173\) 138.724i 0.801873i 0.916106 + 0.400937i \(0.131315\pi\)
−0.916106 + 0.400937i \(0.868685\pi\)
\(174\) 0 0
\(175\) −91.9264 + 70.4284i −0.525294 + 0.402448i
\(176\) 0 0
\(177\) −5.00435 23.7017i −0.0282732 0.133908i
\(178\) 0 0
\(179\) 241.283 1.34795 0.673974 0.738755i \(-0.264586\pi\)
0.673974 + 0.738755i \(0.264586\pi\)
\(180\) 0 0
\(181\) 110.406i 0.609978i 0.952356 + 0.304989i \(0.0986528\pi\)
−0.952356 + 0.304989i \(0.901347\pi\)
\(182\) 0 0
\(183\) 30.9683 6.53861i 0.169226 0.0357301i
\(184\) 0 0
\(185\) 311.871 + 153.943i 1.68579 + 0.832124i
\(186\) 0 0
\(187\) 168.054 0.898684
\(188\) 0 0
\(189\) 101.481 73.1010i 0.536937 0.386778i
\(190\) 0 0
\(191\) 119.468i 0.625487i 0.949838 + 0.312744i \(0.101248\pi\)
−0.949838 + 0.312744i \(0.898752\pi\)
\(192\) 0 0
\(193\) 148.889i 0.771445i −0.922615 0.385722i \(-0.873952\pi\)
0.922615 0.385722i \(-0.126048\pi\)
\(194\) 0 0
\(195\) 12.4878 48.8164i 0.0640399 0.250340i
\(196\) 0 0
\(197\) 277.454i 1.40839i 0.710005 + 0.704197i \(0.248693\pi\)
−0.710005 + 0.704197i \(0.751307\pi\)
\(198\) 0 0
\(199\) 141.225 0.709672 0.354836 0.934929i \(-0.384537\pi\)
0.354836 + 0.934929i \(0.384537\pi\)
\(200\) 0 0
\(201\) −56.9363 269.663i −0.283265 1.34161i
\(202\) 0 0
\(203\) 99.9183i 0.492209i
\(204\) 0 0
\(205\) 138.438 280.459i 0.675305 1.36809i
\(206\) 0 0
\(207\) −297.635 + 131.549i −1.43785 + 0.635501i
\(208\) 0 0
\(209\) 343.302i 1.64259i
\(210\) 0 0
\(211\) 109.167i 0.517381i 0.965960 + 0.258690i \(0.0832909\pi\)
−0.965960 + 0.258690i \(0.916709\pi\)
\(212\) 0 0
\(213\) −232.717 + 49.1356i −1.09257 + 0.230683i
\(214\) 0 0
\(215\) 247.520 + 122.179i 1.15126 + 0.568272i
\(216\) 0 0
\(217\) 131.503i 0.606002i
\(218\) 0 0
\(219\) −173.434 + 36.6185i −0.791934 + 0.167208i
\(220\) 0 0
\(221\) 37.4313 0.169373
\(222\) 0 0
\(223\) 40.5784i 0.181966i 0.995852 + 0.0909830i \(0.0290009\pi\)
−0.995852 + 0.0909830i \(0.970999\pi\)
\(224\) 0 0
\(225\) −197.361 108.045i −0.877160 0.480199i
\(226\) 0 0
\(227\) 439.331i 1.93538i −0.252141 0.967691i \(-0.581135\pi\)
0.252141 0.967691i \(-0.418865\pi\)
\(228\) 0 0
\(229\) 250.730i 1.09489i 0.836841 + 0.547445i \(0.184400\pi\)
−0.836841 + 0.547445i \(0.815600\pi\)
\(230\) 0 0
\(231\) −205.063 + 43.2967i −0.887718 + 0.187432i
\(232\) 0 0
\(233\) 269.289 1.15575 0.577873 0.816127i \(-0.303883\pi\)
0.577873 + 0.816127i \(0.303883\pi\)
\(234\) 0 0
\(235\) 163.764 + 80.8357i 0.696869 + 0.343982i
\(236\) 0 0
\(237\) 53.4967 + 253.372i 0.225724 + 1.06908i
\(238\) 0 0
\(239\) 384.168i 1.60740i 0.595035 + 0.803699i \(0.297138\pi\)
−0.595035 + 0.803699i \(0.702862\pi\)
\(240\) 0 0
\(241\) −250.237 −1.03833 −0.519163 0.854675i \(-0.673756\pi\)
−0.519163 + 0.854675i \(0.673756\pi\)
\(242\) 0 0
\(243\) 209.615 + 122.925i 0.862613 + 0.505864i
\(244\) 0 0
\(245\) −123.490 60.9560i −0.504041 0.248800i
\(246\) 0 0
\(247\) 76.4651i 0.309575i
\(248\) 0 0
\(249\) 29.4108 6.20976i 0.118116 0.0249388i
\(250\) 0 0
\(251\) 319.368 1.27238 0.636191 0.771532i \(-0.280509\pi\)
0.636191 + 0.771532i \(0.280509\pi\)
\(252\) 0 0
\(253\) 545.305 2.15536
\(254\) 0 0
\(255\) 41.4232 161.929i 0.162444 0.635014i
\(256\) 0 0
\(257\) −112.417 −0.437421 −0.218711 0.975790i \(-0.570185\pi\)
−0.218711 + 0.975790i \(0.570185\pi\)
\(258\) 0 0
\(259\) 322.210i 1.24405i
\(260\) 0 0
\(261\) 177.565 78.4801i 0.680324 0.300690i
\(262\) 0 0
\(263\) 65.7040 0.249825 0.124913 0.992168i \(-0.460135\pi\)
0.124913 + 0.992168i \(0.460135\pi\)
\(264\) 0 0
\(265\) −64.8157 + 131.309i −0.244587 + 0.495507i
\(266\) 0 0
\(267\) 52.2631 11.0348i 0.195742 0.0413287i
\(268\) 0 0
\(269\) −162.581 −0.604389 −0.302194 0.953246i \(-0.597719\pi\)
−0.302194 + 0.953246i \(0.597719\pi\)
\(270\) 0 0
\(271\) 351.281 1.29624 0.648119 0.761539i \(-0.275556\pi\)
0.648119 + 0.761539i \(0.275556\pi\)
\(272\) 0 0
\(273\) −45.6745 + 9.64365i −0.167306 + 0.0353247i
\(274\) 0 0
\(275\) 229.306 + 299.301i 0.833839 + 1.08837i
\(276\) 0 0
\(277\) −472.325 −1.70514 −0.852572 0.522610i \(-0.824958\pi\)
−0.852572 + 0.522610i \(0.824958\pi\)
\(278\) 0 0
\(279\) 233.693 103.288i 0.837608 0.370207i
\(280\) 0 0
\(281\) 97.1148i 0.345604i 0.984957 + 0.172802i \(0.0552821\pi\)
−0.984957 + 0.172802i \(0.944718\pi\)
\(282\) 0 0
\(283\) 256.164 0.905173 0.452586 0.891721i \(-0.350501\pi\)
0.452586 + 0.891721i \(0.350501\pi\)
\(284\) 0 0
\(285\) 330.789 + 84.6197i 1.16066 + 0.296911i
\(286\) 0 0
\(287\) −289.756 −1.00960
\(288\) 0 0
\(289\) −164.837 −0.570369
\(290\) 0 0
\(291\) −408.792 + 86.3118i −1.40478 + 0.296604i
\(292\) 0 0
\(293\) 533.766i 1.82173i −0.412708 0.910863i \(-0.635417\pi\)
0.412708 0.910863i \(-0.364583\pi\)
\(294\) 0 0
\(295\) 36.2035 + 17.8704i 0.122724 + 0.0605777i
\(296\) 0 0
\(297\) −238.007 330.409i −0.801372 1.11249i
\(298\) 0 0
\(299\) 121.458 0.406214
\(300\) 0 0
\(301\) 255.726i 0.849587i
\(302\) 0 0
\(303\) 28.9392 + 137.063i 0.0955090 + 0.452352i
\(304\) 0 0
\(305\) −23.3492 + 47.3029i −0.0765548 + 0.155091i
\(306\) 0 0
\(307\) 123.154 0.401152 0.200576 0.979678i \(-0.435719\pi\)
0.200576 + 0.979678i \(0.435719\pi\)
\(308\) 0 0
\(309\) 34.1348 7.20717i 0.110468 0.0233242i
\(310\) 0 0
\(311\) 265.458i 0.853564i −0.904355 0.426782i \(-0.859647\pi\)
0.904355 0.426782i \(-0.140353\pi\)
\(312\) 0 0
\(313\) 372.410i 1.18981i −0.803797 0.594904i \(-0.797190\pi\)
0.803797 0.594904i \(-0.202810\pi\)
\(314\) 0 0
\(315\) −8.82681 + 208.261i −0.0280216 + 0.661145i
\(316\) 0 0
\(317\) 585.780i 1.84789i 0.382530 + 0.923943i \(0.375053\pi\)
−0.382530 + 0.923943i \(0.624947\pi\)
\(318\) 0 0
\(319\) −325.321 −1.01982
\(320\) 0 0
\(321\) 281.199 59.3720i 0.876010 0.184960i
\(322\) 0 0
\(323\) 253.642i 0.785270i
\(324\) 0 0
\(325\) 51.0742 + 66.6645i 0.157151 + 0.205121i
\(326\) 0 0
\(327\) −365.777 + 77.2297i −1.11858 + 0.236176i
\(328\) 0 0
\(329\) 169.193i 0.514265i
\(330\) 0 0
\(331\) 481.587i 1.45495i −0.686136 0.727473i \(-0.740695\pi\)
0.686136 0.727473i \(-0.259305\pi\)
\(332\) 0 0
\(333\) 572.598 253.077i 1.71951 0.759992i
\(334\) 0 0
\(335\) 411.900 + 203.318i 1.22955 + 0.606920i
\(336\) 0 0
\(337\) 581.107i 1.72435i −0.506608 0.862176i \(-0.669101\pi\)
0.506608 0.862176i \(-0.330899\pi\)
\(338\) 0 0
\(339\) 5.75822 + 27.2722i 0.0169859 + 0.0804490i
\(340\) 0 0
\(341\) −428.155 −1.25559
\(342\) 0 0
\(343\) 354.560i 1.03370i
\(344\) 0 0
\(345\) 134.411 525.430i 0.389597 1.52299i
\(346\) 0 0
\(347\) 7.03316i 0.0202685i −0.999949 0.0101342i \(-0.996774\pi\)
0.999949 0.0101342i \(-0.00322588\pi\)
\(348\) 0 0
\(349\) 678.568i 1.94432i −0.234315 0.972161i \(-0.575285\pi\)
0.234315 0.972161i \(-0.424715\pi\)
\(350\) 0 0
\(351\) −53.0124 73.5934i −0.151032 0.209668i
\(352\) 0 0
\(353\) 552.271 1.56451 0.782253 0.622961i \(-0.214070\pi\)
0.782253 + 0.622961i \(0.214070\pi\)
\(354\) 0 0
\(355\) 175.462 355.466i 0.494259 1.00131i
\(356\) 0 0
\(357\) −151.507 + 31.9889i −0.424389 + 0.0896048i
\(358\) 0 0
\(359\) 510.038i 1.42072i −0.703840 0.710359i \(-0.748533\pi\)
0.703840 0.710359i \(-0.251467\pi\)
\(360\) 0 0
\(361\) −157.143 −0.435299
\(362\) 0 0
\(363\) 65.9785 + 312.489i 0.181759 + 0.860851i
\(364\) 0 0
\(365\) 130.764 264.913i 0.358257 0.725789i
\(366\) 0 0
\(367\) 397.074i 1.08194i 0.841040 + 0.540972i \(0.181944\pi\)
−0.841040 + 0.540972i \(0.818056\pi\)
\(368\) 0 0
\(369\) −227.587 514.925i −0.616767 1.39546i
\(370\) 0 0
\(371\) 135.662 0.365667
\(372\) 0 0
\(373\) −192.029 −0.514823 −0.257411 0.966302i \(-0.582870\pi\)
−0.257411 + 0.966302i \(0.582870\pi\)
\(374\) 0 0
\(375\) 344.913 147.174i 0.919767 0.392464i
\(376\) 0 0
\(377\) −72.4601 −0.192202
\(378\) 0 0
\(379\) 150.900i 0.398152i −0.979984 0.199076i \(-0.936206\pi\)
0.979984 0.199076i \(-0.0637941\pi\)
\(380\) 0 0
\(381\) 492.795 104.048i 1.29343 0.273092i
\(382\) 0 0
\(383\) 601.296 1.56996 0.784981 0.619520i \(-0.212673\pi\)
0.784981 + 0.619520i \(0.212673\pi\)
\(384\) 0 0
\(385\) 154.611 313.226i 0.401588 0.813573i
\(386\) 0 0
\(387\) 454.449 200.858i 1.17429 0.519012i
\(388\) 0 0
\(389\) 437.370 1.12434 0.562172 0.827020i \(-0.309966\pi\)
0.562172 + 0.827020i \(0.309966\pi\)
\(390\) 0 0
\(391\) 402.888 1.03040
\(392\) 0 0
\(393\) −3.88963 18.4221i −0.00989727 0.0468757i
\(394\) 0 0
\(395\) −387.016 191.035i −0.979787 0.483634i
\(396\) 0 0
\(397\) 561.277 1.41380 0.706898 0.707315i \(-0.250094\pi\)
0.706898 + 0.707315i \(0.250094\pi\)
\(398\) 0 0
\(399\) −65.3473 309.499i −0.163778 0.775688i
\(400\) 0 0
\(401\) 291.501i 0.726936i −0.931607 0.363468i \(-0.881593\pi\)
0.931607 0.363468i \(-0.118407\pi\)
\(402\) 0 0
\(403\) −95.3648 −0.236637
\(404\) 0 0
\(405\) −377.032 + 147.891i −0.930944 + 0.365162i
\(406\) 0 0
\(407\) −1049.07 −2.57758
\(408\) 0 0
\(409\) 432.727 1.05801 0.529006 0.848618i \(-0.322565\pi\)
0.529006 + 0.848618i \(0.322565\pi\)
\(410\) 0 0
\(411\) −62.5558 296.278i −0.152204 0.720871i
\(412\) 0 0
\(413\) 37.4037i 0.0905658i
\(414\) 0 0
\(415\) −22.1749 + 44.9239i −0.0534335 + 0.108250i
\(416\) 0 0
\(417\) −535.283 + 113.019i −1.28365 + 0.271028i
\(418\) 0 0
\(419\) −133.034 −0.317504 −0.158752 0.987318i \(-0.550747\pi\)
−0.158752 + 0.987318i \(0.550747\pi\)
\(420\) 0 0
\(421\) 691.387i 1.64225i 0.570749 + 0.821124i \(0.306653\pi\)
−0.570749 + 0.821124i \(0.693347\pi\)
\(422\) 0 0
\(423\) 300.672 132.891i 0.710810 0.314164i
\(424\) 0 0
\(425\) 169.418 + 221.133i 0.398631 + 0.520312i
\(426\) 0 0
\(427\) 48.8710 0.114452
\(428\) 0 0
\(429\) 31.3985 + 148.710i 0.0731900 + 0.346644i
\(430\) 0 0
\(431\) 467.222i 1.08404i 0.840365 + 0.542020i \(0.182340\pi\)
−0.840365 + 0.542020i \(0.817660\pi\)
\(432\) 0 0
\(433\) 336.900i 0.778060i 0.921225 + 0.389030i \(0.127190\pi\)
−0.921225 + 0.389030i \(0.872810\pi\)
\(434\) 0 0
\(435\) −80.1876 + 313.464i −0.184339 + 0.720607i
\(436\) 0 0
\(437\) 823.024i 1.88335i
\(438\) 0 0
\(439\) −370.466 −0.843887 −0.421944 0.906622i \(-0.638652\pi\)
−0.421944 + 0.906622i \(0.638652\pi\)
\(440\) 0 0
\(441\) −226.729 + 100.210i −0.514124 + 0.227233i
\(442\) 0 0
\(443\) 126.412i 0.285355i 0.989769 + 0.142677i \(0.0455711\pi\)
−0.989769 + 0.142677i \(0.954429\pi\)
\(444\) 0 0
\(445\) −39.4049 + 79.8299i −0.0885503 + 0.179393i
\(446\) 0 0
\(447\) −128.347 607.880i −0.287130 1.35991i
\(448\) 0 0
\(449\) 372.521i 0.829667i 0.909897 + 0.414834i \(0.136160\pi\)
−0.909897 + 0.414834i \(0.863840\pi\)
\(450\) 0 0
\(451\) 943.410i 2.09182i
\(452\) 0 0
\(453\) −31.8705 150.946i −0.0703544 0.333214i
\(454\) 0 0
\(455\) 34.4372 69.7660i 0.0756862 0.153332i
\(456\) 0 0
\(457\) 46.3235i 0.101364i −0.998715 0.0506822i \(-0.983860\pi\)
0.998715 0.0506822i \(-0.0161396\pi\)
\(458\) 0 0
\(459\) −175.847 244.117i −0.383109 0.531844i
\(460\) 0 0
\(461\) −523.953 −1.13656 −0.568279 0.822836i \(-0.692391\pi\)
−0.568279 + 0.822836i \(0.692391\pi\)
\(462\) 0 0
\(463\) 233.896i 0.505175i 0.967574 + 0.252588i \(0.0812816\pi\)
−0.967574 + 0.252588i \(0.918718\pi\)
\(464\) 0 0
\(465\) −105.535 + 412.550i −0.226957 + 0.887204i
\(466\) 0 0
\(467\) 310.079i 0.663981i −0.943283 0.331990i \(-0.892280\pi\)
0.943283 0.331990i \(-0.107720\pi\)
\(468\) 0 0
\(469\) 425.555i 0.907367i
\(470\) 0 0
\(471\) 89.0680 + 421.846i 0.189104 + 0.895639i
\(472\) 0 0
\(473\) −832.610 −1.76027
\(474\) 0 0
\(475\) −451.732 + 346.089i −0.951014 + 0.728609i
\(476\) 0 0
\(477\) 106.555 + 241.085i 0.223386 + 0.505419i
\(478\) 0 0
\(479\) 618.890i 1.29205i 0.763318 + 0.646023i \(0.223569\pi\)
−0.763318 + 0.646023i \(0.776431\pi\)
\(480\) 0 0
\(481\) −233.665 −0.485789
\(482\) 0 0
\(483\) −491.612 + 103.798i −1.01783 + 0.214904i
\(484\) 0 0
\(485\) 308.217 624.413i 0.635499 1.28745i
\(486\) 0 0
\(487\) 884.108i 1.81542i −0.419602 0.907708i \(-0.637830\pi\)
0.419602 0.907708i \(-0.362170\pi\)
\(488\) 0 0
\(489\) 74.6201 + 353.418i 0.152597 + 0.722736i
\(490\) 0 0
\(491\) 316.866 0.645348 0.322674 0.946510i \(-0.395418\pi\)
0.322674 + 0.946510i \(0.395418\pi\)
\(492\) 0 0
\(493\) −240.357 −0.487540
\(494\) 0 0
\(495\) 678.070 + 28.7390i 1.36984 + 0.0580585i
\(496\) 0 0
\(497\) −367.250 −0.738934
\(498\) 0 0
\(499\) 334.702i 0.670745i −0.942086 0.335373i \(-0.891138\pi\)
0.942086 0.335373i \(-0.108862\pi\)
\(500\) 0 0
\(501\) −72.8155 344.870i −0.145340 0.688364i
\(502\) 0 0
\(503\) 16.1482 0.0321037 0.0160519 0.999871i \(-0.494890\pi\)
0.0160519 + 0.999871i \(0.494890\pi\)
\(504\) 0 0
\(505\) −209.358 103.341i −0.414570 0.204636i
\(506\) 0 0
\(507\) −97.7447 462.941i −0.192790 0.913098i
\(508\) 0 0
\(509\) 442.949 0.870233 0.435117 0.900374i \(-0.356707\pi\)
0.435117 + 0.900374i \(0.356707\pi\)
\(510\) 0 0
\(511\) −273.695 −0.535607
\(512\) 0 0
\(513\) 498.684 359.222i 0.972093 0.700239i
\(514\) 0 0
\(515\) −25.7366 + 52.1395i −0.0499740 + 0.101242i
\(516\) 0 0
\(517\) −550.871 −1.06551
\(518\) 0 0
\(519\) −407.195 + 85.9746i −0.784576 + 0.165654i
\(520\) 0 0
\(521\) 601.758i 1.15501i 0.816389 + 0.577503i \(0.195973\pi\)
−0.816389 + 0.577503i \(0.804027\pi\)
\(522\) 0 0
\(523\) −64.3965 −0.123129 −0.0615645 0.998103i \(-0.519609\pi\)
−0.0615645 + 0.998103i \(0.519609\pi\)
\(524\) 0 0
\(525\) −263.699 226.182i −0.502284 0.430823i
\(526\) 0 0
\(527\) −316.334 −0.600255
\(528\) 0 0
\(529\) 778.301 1.47127
\(530\) 0 0
\(531\) 66.4699 29.3784i 0.125179 0.0553266i
\(532\) 0 0
\(533\) 210.130i 0.394239i
\(534\) 0 0
\(535\) −212.016 + 429.521i −0.396292 + 0.802843i
\(536\) 0 0
\(537\) 149.536 + 708.234i 0.278465 + 1.31887i
\(538\) 0 0
\(539\) 415.397 0.770680
\(540\) 0 0
\(541\) 762.101i 1.40869i 0.709858 + 0.704345i \(0.248759\pi\)
−0.709858 + 0.704345i \(0.751241\pi\)
\(542\) 0 0
\(543\) −324.073 + 68.4244i −0.596820 + 0.126012i
\(544\) 0 0
\(545\) 275.785 558.710i 0.506028 1.02516i
\(546\) 0 0
\(547\) 238.920 0.436783 0.218391 0.975861i \(-0.429919\pi\)
0.218391 + 0.975861i \(0.429919\pi\)
\(548\) 0 0
\(549\) 38.3854 + 86.8485i 0.0699187 + 0.158194i
\(550\) 0 0
\(551\) 491.004i 0.891115i
\(552\) 0 0
\(553\) 399.846i 0.723049i
\(554\) 0 0
\(555\) −258.584 + 1010.84i −0.465917 + 1.82133i
\(556\) 0 0
\(557\) 525.434i 0.943329i 0.881778 + 0.471664i \(0.156347\pi\)
−0.881778 + 0.471664i \(0.843653\pi\)
\(558\) 0 0
\(559\) −185.451 −0.331754
\(560\) 0 0
\(561\) 104.152 + 493.286i 0.185654 + 0.879299i
\(562\) 0 0
\(563\) 465.978i 0.827669i −0.910352 0.413835i \(-0.864189\pi\)
0.910352 0.413835i \(-0.135811\pi\)
\(564\) 0 0
\(565\) −41.6573 20.5625i −0.0737297 0.0363938i
\(566\) 0 0
\(567\) 277.465 + 252.571i 0.489357 + 0.445452i
\(568\) 0 0
\(569\) 217.779i 0.382740i 0.981518 + 0.191370i \(0.0612930\pi\)
−0.981518 + 0.191370i \(0.938707\pi\)
\(570\) 0 0
\(571\) 45.9447i 0.0804636i 0.999190 + 0.0402318i \(0.0128096\pi\)
−0.999190 + 0.0402318i \(0.987190\pi\)
\(572\) 0 0
\(573\) −350.673 + 74.0407i −0.611995 + 0.129216i
\(574\) 0 0
\(575\) 549.732 + 717.536i 0.956056 + 1.24789i
\(576\) 0 0
\(577\) 810.365i 1.40444i 0.711958 + 0.702222i \(0.247809\pi\)
−0.711958 + 0.702222i \(0.752191\pi\)
\(578\) 0 0
\(579\) 437.031 92.2742i 0.754804 0.159368i
\(580\) 0 0
\(581\) 46.4132 0.0798850
\(582\) 0 0
\(583\) 441.699i 0.757632i
\(584\) 0 0
\(585\) 151.029 + 6.40115i 0.258170 + 0.0109421i
\(586\) 0 0
\(587\) 627.562i 1.06910i −0.845137 0.534550i \(-0.820481\pi\)
0.845137 0.534550i \(-0.179519\pi\)
\(588\) 0 0
\(589\) 646.211i 1.09713i
\(590\) 0 0
\(591\) −814.406 + 171.953i −1.37801 + 0.290952i
\(592\) 0 0
\(593\) 913.788 1.54096 0.770479 0.637466i \(-0.220017\pi\)
0.770479 + 0.637466i \(0.220017\pi\)
\(594\) 0 0
\(595\) 114.232 231.421i 0.191986 0.388942i
\(596\) 0 0
\(597\) 87.5244 + 414.535i 0.146607 + 0.694363i
\(598\) 0 0
\(599\) 556.341i 0.928782i −0.885630 0.464391i \(-0.846273\pi\)
0.885630 0.464391i \(-0.153727\pi\)
\(600\) 0 0
\(601\) −265.244 −0.441338 −0.220669 0.975349i \(-0.570824\pi\)
−0.220669 + 0.975349i \(0.570824\pi\)
\(602\) 0 0
\(603\) 756.252 334.249i 1.25415 0.554310i
\(604\) 0 0
\(605\) −477.314 235.608i −0.788949 0.389434i
\(606\) 0 0
\(607\) 496.303i 0.817633i −0.912617 0.408816i \(-0.865942\pi\)
0.912617 0.408816i \(-0.134058\pi\)
\(608\) 0 0
\(609\) 293.289 61.9246i 0.481591 0.101683i
\(610\) 0 0
\(611\) −122.698 −0.200815
\(612\) 0 0
\(613\) −335.215 −0.546843 −0.273421 0.961894i \(-0.588155\pi\)
−0.273421 + 0.961894i \(0.588155\pi\)
\(614\) 0 0
\(615\) 909.024 + 232.539i 1.47809 + 0.378112i
\(616\) 0 0
\(617\) −157.477 −0.255231 −0.127615 0.991824i \(-0.540732\pi\)
−0.127615 + 0.991824i \(0.540732\pi\)
\(618\) 0 0
\(619\) 478.556i 0.773112i 0.922266 + 0.386556i \(0.126335\pi\)
−0.922266 + 0.386556i \(0.873665\pi\)
\(620\) 0 0
\(621\) −570.593 792.115i −0.918829 1.27555i
\(622\) 0 0
\(623\) 82.4763 0.132386
\(624\) 0 0
\(625\) −162.666 + 603.461i −0.260265 + 0.965537i
\(626\) 0 0
\(627\) −1007.69 + 212.762i −1.60716 + 0.339334i
\(628\) 0 0
\(629\) −775.088 −1.23226
\(630\) 0 0
\(631\) 902.124 1.42967 0.714837 0.699291i \(-0.246501\pi\)
0.714837 + 0.699291i \(0.246501\pi\)
\(632\) 0 0
\(633\) −320.437 + 67.6567i −0.506220 + 0.106883i
\(634\) 0 0
\(635\) −371.553 + 752.725i −0.585123 + 1.18539i
\(636\) 0 0
\(637\) 92.5230 0.145248
\(638\) 0 0
\(639\) −288.454 652.639i −0.451415 1.02134i
\(640\) 0 0
\(641\) 354.712i 0.553373i −0.960960 0.276687i \(-0.910764\pi\)
0.960960 0.276687i \(-0.0892364\pi\)
\(642\) 0 0
\(643\) −1223.04 −1.90208 −0.951042 0.309062i \(-0.899985\pi\)
−0.951042 + 0.309062i \(0.899985\pi\)
\(644\) 0 0
\(645\) −205.228 + 802.263i −0.318183 + 1.24382i
\(646\) 0 0
\(647\) 82.6988 0.127819 0.0639094 0.997956i \(-0.479643\pi\)
0.0639094 + 0.997956i \(0.479643\pi\)
\(648\) 0 0
\(649\) −121.782 −0.187645
\(650\) 0 0
\(651\) 385.998 81.4990i 0.592930 0.125191i
\(652\) 0 0
\(653\) 605.030i 0.926539i −0.886217 0.463270i \(-0.846676\pi\)
0.886217 0.463270i \(-0.153324\pi\)
\(654\) 0 0
\(655\) 28.1391 + 13.8898i 0.0429604 + 0.0212057i
\(656\) 0 0
\(657\) −214.972 486.383i −0.327202 0.740309i
\(658\) 0 0
\(659\) −132.704 −0.201372 −0.100686 0.994918i \(-0.532104\pi\)
−0.100686 + 0.994918i \(0.532104\pi\)
\(660\) 0 0
\(661\) 412.823i 0.624544i 0.949993 + 0.312272i \(0.101090\pi\)
−0.949993 + 0.312272i \(0.898910\pi\)
\(662\) 0 0
\(663\) 23.1982 + 109.872i 0.0349897 + 0.165719i
\(664\) 0 0
\(665\) 472.748 + 233.354i 0.710900 + 0.350908i
\(666\) 0 0
\(667\) −779.917 −1.16929
\(668\) 0 0
\(669\) −119.109 + 25.1486i −0.178041 + 0.0375913i
\(670\) 0 0
\(671\) 159.118i 0.237135i
\(672\) 0 0
\(673\) 813.284i 1.20845i 0.796816 + 0.604223i \(0.206516\pi\)
−0.796816 + 0.604223i \(0.793484\pi\)
\(674\) 0 0
\(675\) 194.827 646.272i 0.288633 0.957440i
\(676\) 0 0
\(677\) 671.082i 0.991258i −0.868534 0.495629i \(-0.834938\pi\)
0.868534 0.495629i \(-0.165062\pi\)
\(678\) 0 0
\(679\) −645.114 −0.950094
\(680\) 0 0
\(681\) 1289.56 272.277i 1.89363 0.399819i
\(682\) 0 0
\(683\) 473.636i 0.693464i 0.937964 + 0.346732i \(0.112709\pi\)
−0.937964 + 0.346732i \(0.887291\pi\)
\(684\) 0 0
\(685\) 452.553 + 223.385i 0.660662 + 0.326110i
\(686\) 0 0
\(687\) −735.964 + 155.391i −1.07127 + 0.226187i
\(688\) 0 0
\(689\) 98.3815i 0.142789i
\(690\) 0 0
\(691\) 1191.67i 1.72456i 0.506434 + 0.862279i \(0.330963\pi\)
−0.506434 + 0.862279i \(0.669037\pi\)
\(692\) 0 0
\(693\) −254.176 575.085i −0.366777 0.829848i
\(694\) 0 0
\(695\) 403.588 817.623i 0.580702 1.17644i
\(696\) 0 0
\(697\) 697.020i 1.00003i
\(698\) 0 0
\(699\) 166.892 + 790.439i 0.238759 + 1.13081i
\(700\) 0 0
\(701\) 837.233 1.19434 0.597171 0.802114i \(-0.296291\pi\)
0.597171 + 0.802114i \(0.296291\pi\)
\(702\) 0 0
\(703\) 1583.36i 2.25229i
\(704\) 0 0
\(705\) −135.783 + 530.793i −0.192600 + 0.752898i
\(706\) 0 0
\(707\) 216.298i 0.305938i
\(708\) 0 0
\(709\) 119.868i 0.169067i −0.996421 0.0845334i \(-0.973060\pi\)
0.996421 0.0845334i \(-0.0269400\pi\)
\(710\) 0 0
\(711\) −710.565 + 314.056i −0.999388 + 0.441710i
\(712\) 0 0
\(713\) −1026.45 −1.43962
\(714\) 0 0
\(715\) −227.149 112.123i −0.317691 0.156816i
\(716\) 0 0
\(717\) −1127.64 + 238.089i −1.57273 + 0.332063i
\(718\) 0 0
\(719\) 705.188i 0.980790i −0.871500 0.490395i \(-0.836853\pi\)
0.871500 0.490395i \(-0.163147\pi\)
\(720\) 0 0
\(721\) 53.8680 0.0747129
\(722\) 0 0
\(723\) −155.085 734.516i −0.214502 1.01593i
\(724\) 0 0
\(725\) −327.962 428.072i −0.452362 0.590444i
\(726\) 0 0
\(727\) 503.263i 0.692246i 0.938189 + 0.346123i \(0.112502\pi\)
−0.938189 + 0.346123i \(0.887498\pi\)
\(728\) 0 0
\(729\) −230.911 + 691.463i −0.316750 + 0.948509i
\(730\) 0 0
\(731\) −615.158 −0.841529
\(732\) 0 0
\(733\) −48.5847 −0.0662820 −0.0331410 0.999451i \(-0.510551\pi\)
−0.0331410 + 0.999451i \(0.510551\pi\)
\(734\) 0 0
\(735\) 102.390 400.256i 0.139306 0.544566i
\(736\) 0 0
\(737\) −1385.55 −1.87999
\(738\) 0 0
\(739\) 281.321i 0.380678i −0.981718 0.190339i \(-0.939041\pi\)
0.981718 0.190339i \(-0.0609587\pi\)
\(740\) 0 0
\(741\) −224.447 + 47.3894i −0.302897 + 0.0639534i
\(742\) 0 0
\(743\) −237.515 −0.319671 −0.159835 0.987144i \(-0.551096\pi\)
−0.159835 + 0.987144i \(0.551096\pi\)
\(744\) 0 0
\(745\) 928.513 + 458.324i 1.24633 + 0.615200i
\(746\) 0 0
\(747\) 36.4549 + 82.4807i 0.0488017 + 0.110416i
\(748\) 0 0
\(749\) 443.760 0.592470
\(750\) 0 0
\(751\) 1417.74 1.88780 0.943902 0.330225i \(-0.107125\pi\)
0.943902 + 0.330225i \(0.107125\pi\)
\(752\) 0 0
\(753\) 197.929 + 937.436i 0.262854 + 1.24493i
\(754\) 0 0
\(755\) 230.564 + 113.809i 0.305383 + 0.150740i
\(756\) 0 0
\(757\) −685.674 −0.905778 −0.452889 0.891567i \(-0.649607\pi\)
−0.452889 + 0.891567i \(0.649607\pi\)
\(758\) 0 0
\(759\) 337.954 + 1600.63i 0.445263 + 2.10886i
\(760\) 0 0
\(761\) 1255.49i 1.64980i −0.565282 0.824898i \(-0.691233\pi\)
0.565282 0.824898i \(-0.308767\pi\)
\(762\) 0 0
\(763\) −577.232 −0.756529
\(764\) 0 0
\(765\) 500.979 + 21.2332i 0.654875 + 0.0277559i
\(766\) 0 0
\(767\) −27.1249 −0.0353649
\(768\) 0 0
\(769\) −111.595 −0.145117 −0.0725583 0.997364i \(-0.523116\pi\)
−0.0725583 + 0.997364i \(0.523116\pi\)
\(770\) 0 0
\(771\) −69.6709 329.977i −0.0903643 0.427985i
\(772\) 0 0
\(773\) 496.659i 0.642509i 0.946993 + 0.321254i \(0.104104\pi\)
−0.946993 + 0.321254i \(0.895896\pi\)
\(774\) 0 0
\(775\) −431.631 563.385i −0.556943 0.726949i
\(776\) 0 0
\(777\) 945.779 199.690i 1.21722 0.257002i
\(778\) 0 0
\(779\) −1423.88 −1.82783
\(780\) 0 0
\(781\) 1195.72i 1.53101i
\(782\) 0 0
\(783\) 340.408 + 472.564i 0.434748 + 0.603531i
\(784\) 0 0
\(785\) −644.353 318.060i −0.820832 0.405172i
\(786\) 0 0
\(787\) 122.847 0.156096 0.0780478 0.996950i \(-0.475131\pi\)
0.0780478 + 0.996950i \(0.475131\pi\)
\(788\) 0 0
\(789\) 40.7202 + 192.860i 0.0516099 + 0.244436i
\(790\) 0 0
\(791\) 43.0382i 0.0544099i
\(792\) 0 0
\(793\) 35.4410i 0.0446922i
\(794\) 0 0
\(795\) −425.600 108.873i −0.535346 0.136948i
\(796\) 0 0
\(797\) 159.677i 0.200348i −0.994970 0.100174i \(-0.968060\pi\)
0.994970 0.100174i \(-0.0319399\pi\)
\(798\) 0 0
\(799\) −407.001 −0.509387
\(800\) 0 0
\(801\) 64.7804 + 146.568i 0.0808744 + 0.182982i
\(802\) 0 0
\(803\) 891.117i 1.10973i
\(804\) 0 0
\(805\) 370.662 750.919i 0.460449 0.932819i
\(806\) 0 0
\(807\) −100.760 477.220i −0.124857 0.591351i
\(808\) 0 0
\(809\) 1458.65i 1.80303i −0.432753 0.901513i \(-0.642458\pi\)
0.432753 0.901513i \(-0.357542\pi\)
\(810\) 0 0
\(811\) 607.268i 0.748789i −0.927270 0.374394i \(-0.877851\pi\)
0.927270 0.374394i \(-0.122149\pi\)
\(812\) 0 0
\(813\) 217.707 + 1031.11i 0.267782 + 1.26828i
\(814\) 0 0
\(815\) −539.832 266.467i −0.662370 0.326953i
\(816\) 0 0
\(817\) 1256.65i 1.53813i
\(818\) 0 0
\(819\) −56.6137 128.091i −0.0691255 0.156399i
\(820\) 0 0
\(821\) 706.253 0.860236 0.430118 0.902773i \(-0.358472\pi\)
0.430118 + 0.902773i \(0.358472\pi\)
\(822\) 0 0
\(823\) 101.874i 0.123784i −0.998083 0.0618920i \(-0.980287\pi\)
0.998083 0.0618920i \(-0.0197134\pi\)
\(824\) 0 0
\(825\) −736.421 + 858.571i −0.892631 + 1.04069i
\(826\) 0 0
\(827\) 565.412i 0.683691i −0.939756 0.341845i \(-0.888948\pi\)
0.939756 0.341845i \(-0.111052\pi\)
\(828\) 0 0
\(829\) 1107.57i 1.33603i 0.744146 + 0.668017i \(0.232857\pi\)
−0.744146 + 0.668017i \(0.767143\pi\)
\(830\) 0 0
\(831\) −292.724 1386.41i −0.352256 1.66836i
\(832\) 0 0
\(833\) 306.908 0.368437
\(834\) 0 0
\(835\) 526.776 + 260.022i 0.630870 + 0.311404i
\(836\) 0 0
\(837\) 448.010 + 621.942i 0.535257 + 0.743061i
\(838\) 0 0
\(839\) 220.914i 0.263307i 0.991296 + 0.131653i \(0.0420286\pi\)
−0.991296 + 0.131653i \(0.957971\pi\)
\(840\) 0 0
\(841\) −375.713 −0.446745
\(842\) 0 0
\(843\) −285.060 + 60.1871i −0.338149 + 0.0713964i
\(844\) 0 0
\(845\) 707.124 + 349.044i 0.836833 + 0.413070i
\(846\) 0 0
\(847\) 493.138i 0.582217i
\(848\) 0 0
\(849\) 158.758 + 751.914i 0.186994 + 0.885647i
\(850\) 0 0
\(851\) −2515.03 −2.95538
\(852\) 0 0
\(853\) 666.300 0.781125 0.390562 0.920576i \(-0.372281\pi\)
0.390562 + 0.920576i \(0.372281\pi\)
\(854\) 0 0
\(855\) −43.3755 + 1023.41i −0.0507315 + 1.19697i
\(856\) 0 0
\(857\) 696.570 0.812800 0.406400 0.913695i \(-0.366784\pi\)
0.406400 + 0.913695i \(0.366784\pi\)
\(858\) 0 0
\(859\) 179.165i 0.208574i −0.994547 0.104287i \(-0.966744\pi\)
0.994547 0.104287i \(-0.0332560\pi\)
\(860\) 0 0
\(861\) −179.577 850.518i −0.208568 0.987826i
\(862\) 0 0
\(863\) 216.695 0.251095 0.125548 0.992088i \(-0.459931\pi\)
0.125548 + 0.992088i \(0.459931\pi\)
\(864\) 0 0
\(865\) 307.013 621.974i 0.354929 0.719045i
\(866\) 0 0
\(867\) −102.158 483.843i −0.117829 0.558065i
\(868\) 0 0
\(869\) 1301.85 1.49810
\(870\) 0 0
\(871\) −308.610 −0.354317
\(872\) 0 0
\(873\) −506.699 1146.43i −0.580412 1.31321i
\(874\) 0 0
\(875\) 568.022 112.324i 0.649168 0.128370i
\(876\) 0 0
\(877\) −364.319 −0.415415 −0.207707 0.978191i \(-0.566600\pi\)
−0.207707 + 0.978191i \(0.566600\pi\)
\(878\) 0 0
\(879\) 1566.76 330.803i 1.78243 0.376340i
\(880\) 0 0
\(881\) 77.6990i 0.0881941i 0.999027 + 0.0440970i \(0.0140411\pi\)
−0.999027 + 0.0440970i \(0.985959\pi\)
\(882\) 0 0
\(883\) 1139.19 1.29014 0.645068 0.764125i \(-0.276829\pi\)
0.645068 + 0.764125i \(0.276829\pi\)
\(884\) 0 0
\(885\) −30.0176 + 117.343i −0.0339182 + 0.132591i
\(886\) 0 0
\(887\) −862.886 −0.972814 −0.486407 0.873732i \(-0.661693\pi\)
−0.486407 + 0.873732i \(0.661693\pi\)
\(888\) 0 0
\(889\) 777.679 0.874780
\(890\) 0 0
\(891\) 822.340 903.392i 0.922941 1.01391i
\(892\) 0 0
\(893\) 831.425i 0.931047i
\(894\) 0 0
\(895\) −1081.80 533.988i −1.20871 0.596635i
\(896\) 0 0
\(897\) 75.2740 + 356.514i 0.0839174 + 0.397452i
\(898\) 0 0
\(899\) 612.365 0.681162
\(900\) 0 0
\(901\) 326.341i 0.362199i
\(902\) 0 0
\(903\) 750.628 158.487i 0.831260 0.175511i
\(904\) 0 0
\(905\) 244.342 495.010i 0.269991 0.546972i
\(906\) 0 0
\(907\) 22.0873 0.0243520 0.0121760 0.999926i \(-0.496124\pi\)
0.0121760 + 0.999926i \(0.496124\pi\)
\(908\) 0 0
\(909\) −384.383 + 169.890i −0.422864 + 0.186898i
\(910\) 0 0
\(911\) 565.531i 0.620781i 0.950609 + 0.310390i \(0.100460\pi\)
−0.950609 + 0.310390i \(0.899540\pi\)
\(912\) 0 0
\(913\) 151.115i 0.165515i
\(914\) 0 0
\(915\) −153.318 39.2205i −0.167561 0.0428640i
\(916\) 0 0
\(917\) 29.0719i 0.0317033i
\(918\) 0 0
\(919\) 105.968 0.115308 0.0576539 0.998337i \(-0.481638\pi\)
0.0576539 + 0.998337i \(0.481638\pi\)
\(920\) 0 0
\(921\) 76.3248 + 361.491i 0.0828716 + 0.392499i
\(922\) 0 0
\(923\) 266.328i 0.288546i
\(924\) 0 0
\(925\) −1057.59 1380.42i −1.14334 1.49234i
\(926\) 0 0
\(927\) 42.3102 + 95.7287i 0.0456421 + 0.103267i
\(928\) 0 0
\(929\) 339.002i 0.364911i −0.983214 0.182455i \(-0.941595\pi\)
0.983214 0.182455i \(-0.0584045\pi\)
\(930\) 0 0
\(931\) 626.954i 0.673420i
\(932\) 0 0
\(933\) 779.196 164.518i 0.835151 0.176333i
\(934\) 0 0
\(935\) −753.476 371.924i −0.805857 0.397780i
\(936\) 0 0
\(937\) 216.693i 0.231263i 0.993292 + 0.115631i \(0.0368891\pi\)
−0.993292 + 0.115631i \(0.963111\pi\)
\(938\) 0 0
\(939\) 1093.13 230.802i 1.16414 0.245795i
\(940\) 0 0
\(941\) 575.135 0.611196 0.305598 0.952161i \(-0.401144\pi\)
0.305598 + 0.952161i \(0.401144\pi\)
\(942\) 0 0
\(943\) 2261.71i 2.39842i
\(944\) 0 0
\(945\) −616.775 + 103.161i −0.652672 + 0.109165i
\(946\) 0 0
\(947\) 881.538i 0.930875i 0.885081 + 0.465437i \(0.154103\pi\)
−0.885081 + 0.465437i \(0.845897\pi\)
\(948\) 0 0
\(949\) 198.482i 0.209149i
\(950\) 0 0
\(951\) −1719.43 + 363.039i −1.80803 + 0.381744i
\(952\) 0 0
\(953\) −563.303 −0.591084 −0.295542 0.955330i \(-0.595500\pi\)
−0.295542 + 0.955330i \(0.595500\pi\)
\(954\) 0 0
\(955\) 264.397 535.639i 0.276856 0.560879i
\(956\) 0 0
\(957\) −201.619 954.911i −0.210678 0.997818i
\(958\) 0 0
\(959\) 467.556i 0.487545i
\(960\) 0 0
\(961\) −155.067 −0.161360
\(962\) 0 0
\(963\) 348.548 + 788.604i 0.361940 + 0.818904i
\(964\) 0 0
\(965\) −329.509 + 667.548i −0.341460 + 0.691760i
\(966\) 0 0
\(967\) 607.946i 0.628693i 0.949308 + 0.314347i \(0.101785\pi\)
−0.949308 + 0.314347i \(0.898215\pi\)
\(968\) 0 0
\(969\) −744.513 + 157.195i −0.768331 + 0.162224i
\(970\) 0 0
\(971\) 219.086 0.225629 0.112815 0.993616i \(-0.464013\pi\)
0.112815 + 0.993616i \(0.464013\pi\)
\(972\) 0 0
\(973\) −844.728 −0.868169
\(974\) 0 0
\(975\) −164.026 + 191.233i −0.168232 + 0.196136i
\(976\) 0 0
\(977\) −905.682 −0.927003 −0.463501 0.886096i \(-0.653407\pi\)
−0.463501 + 0.886096i \(0.653407\pi\)
\(978\) 0 0
\(979\) 268.532i 0.274293i
\(980\) 0 0
\(981\) −453.382 1025.80i −0.462163 1.04566i
\(982\) 0 0
\(983\) −30.0755 −0.0305956 −0.0152978 0.999883i \(-0.504870\pi\)
−0.0152978 + 0.999883i \(0.504870\pi\)
\(984\) 0 0
\(985\) 614.039 1243.97i 0.623390 1.26292i
\(986\) 0 0
\(987\) 496.630 104.858i 0.503172 0.106239i
\(988\) 0 0
\(989\) −1996.08 −2.01828
\(990\) 0 0
\(991\) 379.273 0.382717 0.191358 0.981520i \(-0.438711\pi\)
0.191358 + 0.981520i \(0.438711\pi\)
\(992\) 0 0
\(993\) 1413.60 298.465i 1.42356 0.300569i
\(994\) 0 0
\(995\) −633.186 312.547i −0.636368 0.314118i
\(996\) 0 0
\(997\) 1676.89 1.68194 0.840970 0.541082i \(-0.181985\pi\)
0.840970 + 0.541082i \(0.181985\pi\)
\(998\) 0 0
\(999\) 1097.72 + 1523.89i 1.09882 + 1.52542i
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 960.3.i.b.929.37 yes 64
3.2 odd 2 inner 960.3.i.b.929.36 yes 64
4.3 odd 2 inner 960.3.i.b.929.25 64
5.4 even 2 inner 960.3.i.b.929.26 yes 64
8.3 odd 2 inner 960.3.i.b.929.40 yes 64
8.5 even 2 inner 960.3.i.b.929.28 yes 64
12.11 even 2 inner 960.3.i.b.929.32 yes 64
15.14 odd 2 inner 960.3.i.b.929.31 yes 64
20.19 odd 2 inner 960.3.i.b.929.38 yes 64
24.5 odd 2 inner 960.3.i.b.929.29 yes 64
24.11 even 2 inner 960.3.i.b.929.33 yes 64
40.19 odd 2 inner 960.3.i.b.929.27 yes 64
40.29 even 2 inner 960.3.i.b.929.39 yes 64
60.59 even 2 inner 960.3.i.b.929.35 yes 64
120.29 odd 2 inner 960.3.i.b.929.34 yes 64
120.59 even 2 inner 960.3.i.b.929.30 yes 64
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
960.3.i.b.929.25 64 4.3 odd 2 inner
960.3.i.b.929.26 yes 64 5.4 even 2 inner
960.3.i.b.929.27 yes 64 40.19 odd 2 inner
960.3.i.b.929.28 yes 64 8.5 even 2 inner
960.3.i.b.929.29 yes 64 24.5 odd 2 inner
960.3.i.b.929.30 yes 64 120.59 even 2 inner
960.3.i.b.929.31 yes 64 15.14 odd 2 inner
960.3.i.b.929.32 yes 64 12.11 even 2 inner
960.3.i.b.929.33 yes 64 24.11 even 2 inner
960.3.i.b.929.34 yes 64 120.29 odd 2 inner
960.3.i.b.929.35 yes 64 60.59 even 2 inner
960.3.i.b.929.36 yes 64 3.2 odd 2 inner
960.3.i.b.929.37 yes 64 1.1 even 1 trivial
960.3.i.b.929.38 yes 64 20.19 odd 2 inner
960.3.i.b.929.39 yes 64 40.29 even 2 inner
960.3.i.b.929.40 yes 64 8.3 odd 2 inner