Properties

Label 64.4.e.a.17.2
Level $64$
Weight $4$
Character 64.17
Analytic conductor $3.776$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

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

Newform invariants

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

Embedding invariants

Embedding label 17.2
Root \(0.932438 - 1.76934i\) of defining polynomial
Character \(\chi\) \(=\) 64.17
Dual form 64.4.e.a.49.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.98356 - 1.98356i) q^{3} +(-0.596848 + 0.596848i) q^{5} -29.0828i q^{7} -19.1310i q^{9} +O(q^{10})\) \(q+(-1.98356 - 1.98356i) q^{3} +(-0.596848 + 0.596848i) q^{5} -29.0828i q^{7} -19.1310i q^{9} +(-12.1291 + 12.1291i) q^{11} +(-48.5658 - 48.5658i) q^{13} +2.36777 q^{15} +86.7193 q^{17} +(54.8442 + 54.8442i) q^{19} +(-57.6876 + 57.6876i) q^{21} -70.2145i q^{23} +124.288i q^{25} +(-91.5036 + 91.5036i) q^{27} +(63.4021 + 63.4021i) q^{29} +8.86868 q^{31} +48.1178 q^{33} +(17.3580 + 17.3580i) q^{35} +(-21.7145 + 21.7145i) q^{37} +192.667i q^{39} -153.274i q^{41} +(120.951 - 120.951i) q^{43} +(11.4183 + 11.4183i) q^{45} +99.9792 q^{47} -502.812 q^{49} +(-172.013 - 172.013i) q^{51} +(389.132 - 389.132i) q^{53} -14.4785i q^{55} -217.574i q^{57} +(324.819 - 324.819i) q^{59} +(-0.339194 - 0.339194i) q^{61} -556.383 q^{63} +57.9728 q^{65} +(-565.288 - 565.288i) q^{67} +(-139.275 + 139.275i) q^{69} +419.500i q^{71} +374.833i q^{73} +(246.532 - 246.532i) q^{75} +(352.750 + 352.750i) q^{77} +705.750 q^{79} -153.530 q^{81} +(947.092 + 947.092i) q^{83} +(-51.7582 + 51.7582i) q^{85} -251.524i q^{87} -4.72918i q^{89} +(-1412.43 + 1412.43i) q^{91} +(-17.5916 - 17.5916i) q^{93} -65.4673 q^{95} +379.542 q^{97} +(232.042 + 232.042i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{3} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{3} - 2 q^{5} - 18 q^{11} - 2 q^{13} + 124 q^{15} - 4 q^{17} + 26 q^{19} + 52 q^{21} - 184 q^{27} - 202 q^{29} - 368 q^{31} - 4 q^{33} - 476 q^{35} - 10 q^{37} + 838 q^{43} + 194 q^{45} + 944 q^{47} + 94 q^{49} + 1500 q^{51} - 378 q^{53} - 1706 q^{59} + 910 q^{61} - 2628 q^{63} - 492 q^{65} - 1942 q^{67} + 580 q^{69} + 2954 q^{75} - 268 q^{77} + 4416 q^{79} + 482 q^{81} + 2562 q^{83} - 12 q^{85} - 3332 q^{91} - 2192 q^{93} - 6900 q^{95} - 4 q^{97} - 4958 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.98356 1.98356i −0.381737 0.381737i 0.489991 0.871728i \(-0.337000\pi\)
−0.871728 + 0.489991i \(0.837000\pi\)
\(4\) 0 0
\(5\) −0.596848 + 0.596848i −0.0533837 + 0.0533837i −0.733295 0.679911i \(-0.762018\pi\)
0.679911 + 0.733295i \(0.262018\pi\)
\(6\) 0 0
\(7\) 29.0828i 1.57033i −0.619289 0.785163i \(-0.712579\pi\)
0.619289 0.785163i \(-0.287421\pi\)
\(8\) 0 0
\(9\) 19.1310i 0.708554i
\(10\) 0 0
\(11\) −12.1291 + 12.1291i −0.332461 + 0.332461i −0.853520 0.521059i \(-0.825537\pi\)
0.521059 + 0.853520i \(0.325537\pi\)
\(12\) 0 0
\(13\) −48.5658 48.5658i −1.03613 1.03613i −0.999322 0.0368113i \(-0.988280\pi\)
−0.0368113 0.999322i \(-0.511720\pi\)
\(14\) 0 0
\(15\) 2.36777 0.0407570
\(16\) 0 0
\(17\) 86.7193 1.23721 0.618604 0.785703i \(-0.287699\pi\)
0.618604 + 0.785703i \(0.287699\pi\)
\(18\) 0 0
\(19\) 54.8442 + 54.8442i 0.662217 + 0.662217i 0.955902 0.293685i \(-0.0948817\pi\)
−0.293685 + 0.955902i \(0.594882\pi\)
\(20\) 0 0
\(21\) −57.6876 + 57.6876i −0.599451 + 0.599451i
\(22\) 0 0
\(23\) 70.2145i 0.636554i −0.947998 0.318277i \(-0.896896\pi\)
0.947998 0.318277i \(-0.103104\pi\)
\(24\) 0 0
\(25\) 124.288i 0.994300i
\(26\) 0 0
\(27\) −91.5036 + 91.5036i −0.652218 + 0.652218i
\(28\) 0 0
\(29\) 63.4021 + 63.4021i 0.405982 + 0.405982i 0.880335 0.474353i \(-0.157318\pi\)
−0.474353 + 0.880335i \(0.657318\pi\)
\(30\) 0 0
\(31\) 8.86868 0.0513826 0.0256913 0.999670i \(-0.491821\pi\)
0.0256913 + 0.999670i \(0.491821\pi\)
\(32\) 0 0
\(33\) 48.1178 0.253825
\(34\) 0 0
\(35\) 17.3580 + 17.3580i 0.0838298 + 0.0838298i
\(36\) 0 0
\(37\) −21.7145 + 21.7145i −0.0964820 + 0.0964820i −0.753700 0.657218i \(-0.771733\pi\)
0.657218 + 0.753700i \(0.271733\pi\)
\(38\) 0 0
\(39\) 192.667i 0.791060i
\(40\) 0 0
\(41\) 153.274i 0.583840i −0.956443 0.291920i \(-0.905706\pi\)
0.956443 0.291920i \(-0.0942941\pi\)
\(42\) 0 0
\(43\) 120.951 120.951i 0.428949 0.428949i −0.459322 0.888270i \(-0.651907\pi\)
0.888270 + 0.459322i \(0.151907\pi\)
\(44\) 0 0
\(45\) 11.4183 + 11.4183i 0.0378253 + 0.0378253i
\(46\) 0 0
\(47\) 99.9792 0.310286 0.155143 0.987892i \(-0.450416\pi\)
0.155143 + 0.987892i \(0.450416\pi\)
\(48\) 0 0
\(49\) −502.812 −1.46592
\(50\) 0 0
\(51\) −172.013 172.013i −0.472287 0.472287i
\(52\) 0 0
\(53\) 389.132 389.132i 1.00852 1.00852i 0.00855213 0.999963i \(-0.497278\pi\)
0.999963 0.00855213i \(-0.00272226\pi\)
\(54\) 0 0
\(55\) 14.4785i 0.0354960i
\(56\) 0 0
\(57\) 217.574i 0.505585i
\(58\) 0 0
\(59\) 324.819 324.819i 0.716744 0.716744i −0.251193 0.967937i \(-0.580823\pi\)
0.967937 + 0.251193i \(0.0808229\pi\)
\(60\) 0 0
\(61\) −0.339194 0.339194i −0.000711957 0.000711957i 0.706751 0.707463i \(-0.250160\pi\)
−0.707463 + 0.706751i \(0.750160\pi\)
\(62\) 0 0
\(63\) −556.383 −1.11266
\(64\) 0 0
\(65\) 57.9728 0.110625
\(66\) 0 0
\(67\) −565.288 565.288i −1.03076 1.03076i −0.999512 0.0312478i \(-0.990052\pi\)
−0.0312478 0.999512i \(-0.509948\pi\)
\(68\) 0 0
\(69\) −139.275 + 139.275i −0.242996 + 0.242996i
\(70\) 0 0
\(71\) 419.500i 0.701205i 0.936524 + 0.350602i \(0.114023\pi\)
−0.936524 + 0.350602i \(0.885977\pi\)
\(72\) 0 0
\(73\) 374.833i 0.600971i 0.953786 + 0.300485i \(0.0971487\pi\)
−0.953786 + 0.300485i \(0.902851\pi\)
\(74\) 0 0
\(75\) 246.532 246.532i 0.379561 0.379561i
\(76\) 0 0
\(77\) 352.750 + 352.750i 0.522072 + 0.522072i
\(78\) 0 0
\(79\) 705.750 1.00510 0.502551 0.864547i \(-0.332395\pi\)
0.502551 + 0.864547i \(0.332395\pi\)
\(80\) 0 0
\(81\) −153.530 −0.210604
\(82\) 0 0
\(83\) 947.092 + 947.092i 1.25249 + 1.25249i 0.954599 + 0.297893i \(0.0962839\pi\)
0.297893 + 0.954599i \(0.403716\pi\)
\(84\) 0 0
\(85\) −51.7582 + 51.7582i −0.0660467 + 0.0660467i
\(86\) 0 0
\(87\) 251.524i 0.309956i
\(88\) 0 0
\(89\) 4.72918i 0.00563249i −0.999996 0.00281625i \(-0.999104\pi\)
0.999996 0.00281625i \(-0.000896440\pi\)
\(90\) 0 0
\(91\) −1412.43 + 1412.43i −1.62707 + 1.62707i
\(92\) 0 0
\(93\) −17.5916 17.5916i −0.0196146 0.0196146i
\(94\) 0 0
\(95\) −65.4673 −0.0707032
\(96\) 0 0
\(97\) 379.542 0.397285 0.198643 0.980072i \(-0.436347\pi\)
0.198643 + 0.980072i \(0.436347\pi\)
\(98\) 0 0
\(99\) 232.042 + 232.042i 0.235567 + 0.235567i
\(100\) 0 0
\(101\) −391.005 + 391.005i −0.385212 + 0.385212i −0.872976 0.487764i \(-0.837813\pi\)
0.487764 + 0.872976i \(0.337813\pi\)
\(102\) 0 0
\(103\) 307.935i 0.294580i 0.989093 + 0.147290i \(0.0470551\pi\)
−0.989093 + 0.147290i \(0.952945\pi\)
\(104\) 0 0
\(105\) 68.8615i 0.0640018i
\(106\) 0 0
\(107\) 601.607 601.607i 0.543548 0.543548i −0.381019 0.924567i \(-0.624427\pi\)
0.924567 + 0.381019i \(0.124427\pi\)
\(108\) 0 0
\(109\) −948.890 948.890i −0.833827 0.833827i 0.154211 0.988038i \(-0.450717\pi\)
−0.988038 + 0.154211i \(0.950717\pi\)
\(110\) 0 0
\(111\) 86.1439 0.0736614
\(112\) 0 0
\(113\) 1824.02 1.51849 0.759244 0.650807i \(-0.225569\pi\)
0.759244 + 0.650807i \(0.225569\pi\)
\(114\) 0 0
\(115\) 41.9074 + 41.9074i 0.0339816 + 0.0339816i
\(116\) 0 0
\(117\) −929.111 + 929.111i −0.734157 + 0.734157i
\(118\) 0 0
\(119\) 2522.04i 1.94282i
\(120\) 0 0
\(121\) 1036.77i 0.778939i
\(122\) 0 0
\(123\) −304.029 + 304.029i −0.222873 + 0.222873i
\(124\) 0 0
\(125\) −148.787 148.787i −0.106463 0.106463i
\(126\) 0 0
\(127\) −988.748 −0.690844 −0.345422 0.938447i \(-0.612264\pi\)
−0.345422 + 0.938447i \(0.612264\pi\)
\(128\) 0 0
\(129\) −479.826 −0.327491
\(130\) 0 0
\(131\) −793.572 793.572i −0.529273 0.529273i 0.391083 0.920356i \(-0.372101\pi\)
−0.920356 + 0.391083i \(0.872101\pi\)
\(132\) 0 0
\(133\) 1595.03 1595.03i 1.03990 1.03990i
\(134\) 0 0
\(135\) 109.227i 0.0696356i
\(136\) 0 0
\(137\) 1595.30i 0.994856i −0.867505 0.497428i \(-0.834278\pi\)
0.867505 0.497428i \(-0.165722\pi\)
\(138\) 0 0
\(139\) 277.696 277.696i 0.169452 0.169452i −0.617286 0.786738i \(-0.711768\pi\)
0.786738 + 0.617286i \(0.211768\pi\)
\(140\) 0 0
\(141\) −198.315 198.315i −0.118448 0.118448i
\(142\) 0 0
\(143\) 1178.12 0.688948
\(144\) 0 0
\(145\) −75.6828 −0.0433456
\(146\) 0 0
\(147\) 997.359 + 997.359i 0.559597 + 0.559597i
\(148\) 0 0
\(149\) −593.272 + 593.272i −0.326193 + 0.326193i −0.851137 0.524944i \(-0.824086\pi\)
0.524944 + 0.851137i \(0.324086\pi\)
\(150\) 0 0
\(151\) 160.655i 0.0865821i 0.999063 + 0.0432911i \(0.0137843\pi\)
−0.999063 + 0.0432911i \(0.986216\pi\)
\(152\) 0 0
\(153\) 1659.02i 0.876629i
\(154\) 0 0
\(155\) −5.29325 + 5.29325i −0.00274299 + 0.00274299i
\(156\) 0 0
\(157\) −705.762 705.762i −0.358764 0.358764i 0.504593 0.863357i \(-0.331642\pi\)
−0.863357 + 0.504593i \(0.831642\pi\)
\(158\) 0 0
\(159\) −1543.73 −0.769975
\(160\) 0 0
\(161\) −2042.04 −0.999598
\(162\) 0 0
\(163\) −1872.64 1872.64i −0.899855 0.899855i 0.0955676 0.995423i \(-0.469533\pi\)
−0.995423 + 0.0955676i \(0.969533\pi\)
\(164\) 0 0
\(165\) −28.7190 + 28.7190i −0.0135501 + 0.0135501i
\(166\) 0 0
\(167\) 3852.19i 1.78498i 0.451066 + 0.892490i \(0.351044\pi\)
−0.451066 + 0.892490i \(0.648956\pi\)
\(168\) 0 0
\(169\) 2520.28i 1.14715i
\(170\) 0 0
\(171\) 1049.22 1049.22i 0.469217 0.469217i
\(172\) 0 0
\(173\) 2625.61 + 2625.61i 1.15388 + 1.15388i 0.985768 + 0.168112i \(0.0537671\pi\)
0.168112 + 0.985768i \(0.446233\pi\)
\(174\) 0 0
\(175\) 3614.64 1.56138
\(176\) 0 0
\(177\) −1288.60 −0.547215
\(178\) 0 0
\(179\) 1236.73 + 1236.73i 0.516413 + 0.516413i 0.916484 0.400071i \(-0.131015\pi\)
−0.400071 + 0.916484i \(0.631015\pi\)
\(180\) 0 0
\(181\) 1574.90 1574.90i 0.646748 0.646748i −0.305458 0.952206i \(-0.598809\pi\)
0.952206 + 0.305458i \(0.0988094\pi\)
\(182\) 0 0
\(183\) 1.34563i 0.000543560i
\(184\) 0 0
\(185\) 25.9204i 0.0103011i
\(186\) 0 0
\(187\) −1051.83 + 1051.83i −0.411323 + 0.411323i
\(188\) 0 0
\(189\) 2661.19 + 2661.19i 1.02419 + 1.02419i
\(190\) 0 0
\(191\) −3585.92 −1.35847 −0.679236 0.733920i \(-0.737689\pi\)
−0.679236 + 0.733920i \(0.737689\pi\)
\(192\) 0 0
\(193\) 523.601 0.195283 0.0976415 0.995222i \(-0.468870\pi\)
0.0976415 + 0.995222i \(0.468870\pi\)
\(194\) 0 0
\(195\) −114.993 114.993i −0.0422297 0.0422297i
\(196\) 0 0
\(197\) −1125.64 + 1125.64i −0.407098 + 0.407098i −0.880725 0.473627i \(-0.842944\pi\)
0.473627 + 0.880725i \(0.342944\pi\)
\(198\) 0 0
\(199\) 2312.48i 0.823757i 0.911239 + 0.411878i \(0.135127\pi\)
−0.911239 + 0.411878i \(0.864873\pi\)
\(200\) 0 0
\(201\) 2242.57i 0.786957i
\(202\) 0 0
\(203\) 1843.91 1843.91i 0.637524 0.637524i
\(204\) 0 0
\(205\) 91.4815 + 91.4815i 0.0311675 + 0.0311675i
\(206\) 0 0
\(207\) −1343.27 −0.451033
\(208\) 0 0
\(209\) −1330.43 −0.440323
\(210\) 0 0
\(211\) −1418.59 1418.59i −0.462842 0.462842i 0.436744 0.899586i \(-0.356132\pi\)
−0.899586 + 0.436744i \(0.856132\pi\)
\(212\) 0 0
\(213\) 832.105 832.105i 0.267675 0.267675i
\(214\) 0 0
\(215\) 144.378i 0.0457977i
\(216\) 0 0
\(217\) 257.926i 0.0806875i
\(218\) 0 0
\(219\) 743.504 743.504i 0.229413 0.229413i
\(220\) 0 0
\(221\) −4211.60 4211.60i −1.28191 1.28191i
\(222\) 0 0
\(223\) 4315.08 1.29578 0.647890 0.761734i \(-0.275651\pi\)
0.647890 + 0.761734i \(0.275651\pi\)
\(224\) 0 0
\(225\) 2377.74 0.704516
\(226\) 0 0
\(227\) −701.203 701.203i −0.205024 0.205024i 0.597124 0.802149i \(-0.296310\pi\)
−0.802149 + 0.597124i \(0.796310\pi\)
\(228\) 0 0
\(229\) −663.351 + 663.351i −0.191421 + 0.191421i −0.796310 0.604889i \(-0.793218\pi\)
0.604889 + 0.796310i \(0.293218\pi\)
\(230\) 0 0
\(231\) 1399.40i 0.398588i
\(232\) 0 0
\(233\) 3490.15i 0.981318i −0.871352 0.490659i \(-0.836756\pi\)
0.871352 0.490659i \(-0.163244\pi\)
\(234\) 0 0
\(235\) −59.6724 + 59.6724i −0.0165642 + 0.0165642i
\(236\) 0 0
\(237\) −1399.90 1399.90i −0.383684 0.383684i
\(238\) 0 0
\(239\) −2950.43 −0.798525 −0.399263 0.916837i \(-0.630734\pi\)
−0.399263 + 0.916837i \(0.630734\pi\)
\(240\) 0 0
\(241\) −1128.96 −0.301755 −0.150877 0.988552i \(-0.548210\pi\)
−0.150877 + 0.988552i \(0.548210\pi\)
\(242\) 0 0
\(243\) 2775.13 + 2775.13i 0.732613 + 0.732613i
\(244\) 0 0
\(245\) 300.102 300.102i 0.0782565 0.0782565i
\(246\) 0 0
\(247\) 5327.11i 1.37229i
\(248\) 0 0
\(249\) 3757.23i 0.956244i
\(250\) 0 0
\(251\) −4621.86 + 4621.86i −1.16227 + 1.16227i −0.178291 + 0.983978i \(0.557057\pi\)
−0.983978 + 0.178291i \(0.942943\pi\)
\(252\) 0 0
\(253\) 851.642 + 851.642i 0.211630 + 0.211630i
\(254\) 0 0
\(255\) 205.331 0.0504249
\(256\) 0 0
\(257\) 610.977 0.148295 0.0741473 0.997247i \(-0.476377\pi\)
0.0741473 + 0.997247i \(0.476377\pi\)
\(258\) 0 0
\(259\) 631.518 + 631.518i 0.151508 + 0.151508i
\(260\) 0 0
\(261\) 1212.94 1212.94i 0.287660 0.287660i
\(262\) 0 0
\(263\) 4973.57i 1.16610i −0.812438 0.583048i \(-0.801860\pi\)
0.812438 0.583048i \(-0.198140\pi\)
\(264\) 0 0
\(265\) 464.505i 0.107677i
\(266\) 0 0
\(267\) −9.38061 + 9.38061i −0.00215013 + 0.00215013i
\(268\) 0 0
\(269\) −938.415 938.415i −0.212700 0.212700i 0.592714 0.805413i \(-0.298057\pi\)
−0.805413 + 0.592714i \(0.798057\pi\)
\(270\) 0 0
\(271\) 4010.64 0.898999 0.449500 0.893280i \(-0.351602\pi\)
0.449500 + 0.893280i \(0.351602\pi\)
\(272\) 0 0
\(273\) 5603.29 1.24222
\(274\) 0 0
\(275\) −1507.50 1507.50i −0.330566 0.330566i
\(276\) 0 0
\(277\) −3534.99 + 3534.99i −0.766776 + 0.766776i −0.977538 0.210762i \(-0.932406\pi\)
0.210762 + 0.977538i \(0.432406\pi\)
\(278\) 0 0
\(279\) 169.666i 0.0364074i
\(280\) 0 0
\(281\) 7468.35i 1.58550i 0.609550 + 0.792748i \(0.291350\pi\)
−0.609550 + 0.792748i \(0.708650\pi\)
\(282\) 0 0
\(283\) 2249.22 2249.22i 0.472447 0.472447i −0.430259 0.902705i \(-0.641578\pi\)
0.902705 + 0.430259i \(0.141578\pi\)
\(284\) 0 0
\(285\) 129.858 + 129.858i 0.0269900 + 0.0269900i
\(286\) 0 0
\(287\) −4457.66 −0.916819
\(288\) 0 0
\(289\) 2607.24 0.530682
\(290\) 0 0
\(291\) −752.845 752.845i −0.151658 0.151658i
\(292\) 0 0
\(293\) −3952.79 + 3952.79i −0.788139 + 0.788139i −0.981189 0.193050i \(-0.938162\pi\)
0.193050 + 0.981189i \(0.438162\pi\)
\(294\) 0 0
\(295\) 387.736i 0.0765249i
\(296\) 0 0
\(297\) 2219.72i 0.433674i
\(298\) 0 0
\(299\) −3410.03 + 3410.03i −0.659555 + 0.659555i
\(300\) 0 0
\(301\) −3517.59 3517.59i −0.673589 0.673589i
\(302\) 0 0
\(303\) 1551.16 0.294099
\(304\) 0 0
\(305\) 0.404895 7.60138e−5
\(306\) 0 0
\(307\) 3855.24 + 3855.24i 0.716711 + 0.716711i 0.967930 0.251219i \(-0.0808316\pi\)
−0.251219 + 0.967930i \(0.580832\pi\)
\(308\) 0 0
\(309\) 610.809 610.809i 0.112452 0.112452i
\(310\) 0 0
\(311\) 5194.39i 0.947096i 0.880768 + 0.473548i \(0.157027\pi\)
−0.880768 + 0.473548i \(0.842973\pi\)
\(312\) 0 0
\(313\) 4710.01i 0.850561i 0.905062 + 0.425281i \(0.139825\pi\)
−0.905062 + 0.425281i \(0.860175\pi\)
\(314\) 0 0
\(315\) 332.076 332.076i 0.0593980 0.0593980i
\(316\) 0 0
\(317\) 5680.21 + 5680.21i 1.00641 + 1.00641i 0.999979 + 0.00643263i \(0.00204758\pi\)
0.00643263 + 0.999979i \(0.497952\pi\)
\(318\) 0 0
\(319\) −1538.03 −0.269946
\(320\) 0 0
\(321\) −2386.65 −0.414984
\(322\) 0 0
\(323\) 4756.05 + 4756.05i 0.819300 + 0.819300i
\(324\) 0 0
\(325\) 6036.13 6036.13i 1.03023 1.03023i
\(326\) 0 0
\(327\) 3764.36i 0.636605i
\(328\) 0 0
\(329\) 2907.68i 0.487251i
\(330\) 0 0
\(331\) −1815.80 + 1815.80i −0.301526 + 0.301526i −0.841611 0.540084i \(-0.818392\pi\)
0.540084 + 0.841611i \(0.318392\pi\)
\(332\) 0 0
\(333\) 415.418 + 415.418i 0.0683627 + 0.0683627i
\(334\) 0 0
\(335\) 674.782 0.110052
\(336\) 0 0
\(337\) −2683.29 −0.433733 −0.216867 0.976201i \(-0.569584\pi\)
−0.216867 + 0.976201i \(0.569584\pi\)
\(338\) 0 0
\(339\) −3618.05 3618.05i −0.579662 0.579662i
\(340\) 0 0
\(341\) −107.569 + 107.569i −0.0170827 + 0.0170827i
\(342\) 0 0
\(343\) 4647.79i 0.731653i
\(344\) 0 0
\(345\) 166.252i 0.0259440i
\(346\) 0 0
\(347\) 5291.81 5291.81i 0.818671 0.818671i −0.167244 0.985916i \(-0.553487\pi\)
0.985916 + 0.167244i \(0.0534868\pi\)
\(348\) 0 0
\(349\) 73.7084 + 73.7084i 0.0113052 + 0.0113052i 0.712737 0.701432i \(-0.247455\pi\)
−0.701432 + 0.712737i \(0.747455\pi\)
\(350\) 0 0
\(351\) 8887.90 1.35157
\(352\) 0 0
\(353\) −5067.25 −0.764030 −0.382015 0.924156i \(-0.624770\pi\)
−0.382015 + 0.924156i \(0.624770\pi\)
\(354\) 0 0
\(355\) −250.378 250.378i −0.0374329 0.0374329i
\(356\) 0 0
\(357\) −5002.63 + 5002.63i −0.741645 + 0.741645i
\(358\) 0 0
\(359\) 970.230i 0.142637i −0.997454 0.0713186i \(-0.977279\pi\)
0.997454 0.0713186i \(-0.0227207\pi\)
\(360\) 0 0
\(361\) 843.224i 0.122937i
\(362\) 0 0
\(363\) 2056.49 2056.49i 0.297350 0.297350i
\(364\) 0 0
\(365\) −223.718 223.718i −0.0320821 0.0320821i
\(366\) 0 0
\(367\) −13451.4 −1.91323 −0.956617 0.291347i \(-0.905896\pi\)
−0.956617 + 0.291347i \(0.905896\pi\)
\(368\) 0 0
\(369\) −2932.29 −0.413682
\(370\) 0 0
\(371\) −11317.1 11317.1i −1.58370 1.58370i
\(372\) 0 0
\(373\) 5898.22 5898.22i 0.818762 0.818762i −0.167167 0.985929i \(-0.553462\pi\)
0.985929 + 0.167167i \(0.0534619\pi\)
\(374\) 0 0
\(375\) 590.255i 0.0812817i
\(376\) 0 0
\(377\) 6158.35i 0.841302i
\(378\) 0 0
\(379\) 4446.72 4446.72i 0.602673 0.602673i −0.338348 0.941021i \(-0.609868\pi\)
0.941021 + 0.338348i \(0.109868\pi\)
\(380\) 0 0
\(381\) 1961.24 + 1961.24i 0.263721 + 0.263721i
\(382\) 0 0
\(383\) 6417.68 0.856209 0.428105 0.903729i \(-0.359181\pi\)
0.428105 + 0.903729i \(0.359181\pi\)
\(384\) 0 0
\(385\) −421.076 −0.0557403
\(386\) 0 0
\(387\) −2313.90 2313.90i −0.303933 0.303933i
\(388\) 0 0
\(389\) 6555.61 6555.61i 0.854455 0.854455i −0.136223 0.990678i \(-0.543497\pi\)
0.990678 + 0.136223i \(0.0434965\pi\)
\(390\) 0 0
\(391\) 6088.96i 0.787549i
\(392\) 0 0
\(393\) 3148.20i 0.404086i
\(394\) 0 0
\(395\) −421.226 + 421.226i −0.0536561 + 0.0536561i
\(396\) 0 0
\(397\) 8902.51 + 8902.51i 1.12545 + 1.12545i 0.990908 + 0.134543i \(0.0429566\pi\)
0.134543 + 0.990908i \(0.457043\pi\)
\(398\) 0 0
\(399\) −6327.66 −0.793933
\(400\) 0 0
\(401\) −6425.77 −0.800218 −0.400109 0.916468i \(-0.631028\pi\)
−0.400109 + 0.916468i \(0.631028\pi\)
\(402\) 0 0
\(403\) −430.715 430.715i −0.0532393 0.0532393i
\(404\) 0 0
\(405\) 91.6341 91.6341i 0.0112428 0.0112428i
\(406\) 0 0
\(407\) 526.755i 0.0641530i
\(408\) 0 0
\(409\) 12796.0i 1.54699i −0.633801 0.773496i \(-0.718506\pi\)
0.633801 0.773496i \(-0.281494\pi\)
\(410\) 0 0
\(411\) −3164.37 + 3164.37i −0.379773 + 0.379773i
\(412\) 0 0
\(413\) −9446.67 9446.67i −1.12552 1.12552i
\(414\) 0 0
\(415\) −1130.54 −0.133725
\(416\) 0 0
\(417\) −1101.65 −0.129372
\(418\) 0 0
\(419\) 6545.21 + 6545.21i 0.763137 + 0.763137i 0.976888 0.213751i \(-0.0685681\pi\)
−0.213751 + 0.976888i \(0.568568\pi\)
\(420\) 0 0
\(421\) −6390.00 + 6390.00i −0.739738 + 0.739738i −0.972527 0.232789i \(-0.925215\pi\)
0.232789 + 0.972527i \(0.425215\pi\)
\(422\) 0 0
\(423\) 1912.70i 0.219855i
\(424\) 0 0
\(425\) 10778.1i 1.23016i
\(426\) 0 0
\(427\) −9.86474 + 9.86474i −0.00111801 + 0.00111801i
\(428\) 0 0
\(429\) −2336.88 2336.88i −0.262997 0.262997i
\(430\) 0 0
\(431\) 10639.3 1.18904 0.594519 0.804081i \(-0.297342\pi\)
0.594519 + 0.804081i \(0.297342\pi\)
\(432\) 0 0
\(433\) 3806.14 0.422428 0.211214 0.977440i \(-0.432258\pi\)
0.211214 + 0.977440i \(0.432258\pi\)
\(434\) 0 0
\(435\) 150.121 + 150.121i 0.0165466 + 0.0165466i
\(436\) 0 0
\(437\) 3850.86 3850.86i 0.421537 0.421537i
\(438\) 0 0
\(439\) 14102.8i 1.53323i 0.642106 + 0.766616i \(0.278061\pi\)
−0.642106 + 0.766616i \(0.721939\pi\)
\(440\) 0 0
\(441\) 9619.28i 1.03869i
\(442\) 0 0
\(443\) 7662.45 7662.45i 0.821792 0.821792i −0.164573 0.986365i \(-0.552625\pi\)
0.986365 + 0.164573i \(0.0526246\pi\)
\(444\) 0 0
\(445\) 2.82260 + 2.82260i 0.000300683 + 0.000300683i
\(446\) 0 0
\(447\) 2353.58 0.249040
\(448\) 0 0
\(449\) 13679.4 1.43779 0.718897 0.695117i \(-0.244647\pi\)
0.718897 + 0.695117i \(0.244647\pi\)
\(450\) 0 0
\(451\) 1859.09 + 1859.09i 0.194104 + 0.194104i
\(452\) 0 0
\(453\) 318.669 318.669i 0.0330516 0.0330516i
\(454\) 0 0
\(455\) 1686.01i 0.173718i
\(456\) 0 0
\(457\) 7913.48i 0.810016i −0.914313 0.405008i \(-0.867269\pi\)
0.914313 0.405008i \(-0.132731\pi\)
\(458\) 0 0
\(459\) −7935.13 + 7935.13i −0.806929 + 0.806929i
\(460\) 0 0
\(461\) −580.215 580.215i −0.0586189 0.0586189i 0.677190 0.735809i \(-0.263198\pi\)
−0.735809 + 0.677190i \(0.763198\pi\)
\(462\) 0 0
\(463\) −14236.5 −1.42899 −0.714497 0.699638i \(-0.753344\pi\)
−0.714497 + 0.699638i \(0.753344\pi\)
\(464\) 0 0
\(465\) 20.9990 0.00209420
\(466\) 0 0
\(467\) 8344.57 + 8344.57i 0.826853 + 0.826853i 0.987080 0.160227i \(-0.0512225\pi\)
−0.160227 + 0.987080i \(0.551223\pi\)
\(468\) 0 0
\(469\) −16440.2 + 16440.2i −1.61863 + 1.61863i
\(470\) 0 0
\(471\) 2799.85i 0.273907i
\(472\) 0 0
\(473\) 2934.05i 0.285217i
\(474\) 0 0
\(475\) −6816.45 + 6816.45i −0.658443 + 0.658443i
\(476\) 0 0
\(477\) −7444.46 7444.46i −0.714588 0.714588i
\(478\) 0 0
\(479\) 5563.77 0.530720 0.265360 0.964149i \(-0.414509\pi\)
0.265360 + 0.964149i \(0.414509\pi\)
\(480\) 0 0
\(481\) 2109.16 0.199936
\(482\) 0 0
\(483\) 4050.51 + 4050.51i 0.381583 + 0.381583i
\(484\) 0 0
\(485\) −226.529 + 226.529i −0.0212085 + 0.0212085i
\(486\) 0 0
\(487\) 18150.5i 1.68886i −0.535662 0.844432i \(-0.679938\pi\)
0.535662 0.844432i \(-0.320062\pi\)
\(488\) 0 0
\(489\) 7428.99i 0.687015i
\(490\) 0 0
\(491\) −11593.0 + 11593.0i −1.06555 + 1.06555i −0.0678570 + 0.997695i \(0.521616\pi\)
−0.997695 + 0.0678570i \(0.978384\pi\)
\(492\) 0 0
\(493\) 5498.18 + 5498.18i 0.502284 + 0.502284i
\(494\) 0 0
\(495\) −276.988 −0.0251509
\(496\) 0 0
\(497\) 12200.3 1.10112
\(498\) 0 0
\(499\) −3109.58 3109.58i −0.278966 0.278966i 0.553730 0.832696i \(-0.313204\pi\)
−0.832696 + 0.553730i \(0.813204\pi\)
\(500\) 0 0
\(501\) 7641.07 7641.07i 0.681393 0.681393i
\(502\) 0 0
\(503\) 6221.21i 0.551471i −0.961233 0.275736i \(-0.911079\pi\)
0.961233 0.275736i \(-0.0889215\pi\)
\(504\) 0 0
\(505\) 466.740i 0.0411281i
\(506\) 0 0
\(507\) 4999.13 4999.13i 0.437907 0.437907i
\(508\) 0 0
\(509\) 13800.4 + 13800.4i 1.20176 + 1.20176i 0.973632 + 0.228124i \(0.0732591\pi\)
0.228124 + 0.973632i \(0.426741\pi\)
\(510\) 0 0
\(511\) 10901.2 0.943720
\(512\) 0 0
\(513\) −10036.9 −0.863820
\(514\) 0 0
\(515\) −183.791 183.791i −0.0157258 0.0157258i
\(516\) 0 0
\(517\) −1212.66 + 1212.66i −0.103158 + 0.103158i
\(518\) 0 0
\(519\) 10416.1i 0.880957i
\(520\) 0 0
\(521\) 6874.63i 0.578086i 0.957316 + 0.289043i \(0.0933371\pi\)
−0.957316 + 0.289043i \(0.906663\pi\)
\(522\) 0 0
\(523\) −2306.52 + 2306.52i −0.192843 + 0.192843i −0.796924 0.604080i \(-0.793541\pi\)
0.604080 + 0.796924i \(0.293541\pi\)
\(524\) 0 0
\(525\) −7169.85 7169.85i −0.596034 0.596034i
\(526\) 0 0
\(527\) 769.086 0.0635710
\(528\) 0 0
\(529\) 7236.92 0.594799
\(530\) 0 0
\(531\) −6214.11 6214.11i −0.507852 0.507852i
\(532\) 0 0
\(533\) −7443.90 + 7443.90i −0.604936 + 0.604936i
\(534\) 0 0
\(535\) 718.136i 0.0580332i
\(536\) 0 0
\(537\) 4906.28i 0.394267i
\(538\) 0 0
\(539\) 6098.68 6098.68i 0.487363 0.487363i
\(540\) 0 0
\(541\) −13240.0 13240.0i −1.05218 1.05218i −0.998561 0.0536210i \(-0.982924\pi\)
−0.0536210 0.998561i \(-0.517076\pi\)
\(542\) 0 0
\(543\) −6247.82 −0.493775
\(544\) 0 0
\(545\) 1132.69 0.0890256
\(546\) 0 0
\(547\) 13271.3 + 13271.3i 1.03737 + 1.03737i 0.999274 + 0.0380940i \(0.0121286\pi\)
0.0380940 + 0.999274i \(0.487871\pi\)
\(548\) 0 0
\(549\) −6.48912 + 6.48912i −0.000504460 + 0.000504460i
\(550\) 0 0
\(551\) 6954.47i 0.537696i
\(552\) 0 0
\(553\) 20525.2i 1.57834i
\(554\) 0 0
\(555\) −51.4148 + 51.4148i −0.00393232 + 0.00393232i
\(556\) 0 0
\(557\) −8500.61 8500.61i −0.646647 0.646647i 0.305534 0.952181i \(-0.401165\pi\)
−0.952181 + 0.305534i \(0.901165\pi\)
\(558\) 0 0
\(559\) −11748.1 −0.888896
\(560\) 0 0
\(561\) 4172.74 0.314034
\(562\) 0 0
\(563\) −17327.2 17327.2i −1.29708 1.29708i −0.930314 0.366763i \(-0.880466\pi\)
−0.366763 0.930314i \(-0.619534\pi\)
\(564\) 0 0
\(565\) −1088.66 + 1088.66i −0.0810625 + 0.0810625i
\(566\) 0 0
\(567\) 4465.09i 0.330716i
\(568\) 0 0
\(569\) 8998.54i 0.662985i −0.943458 0.331492i \(-0.892448\pi\)
0.943458 0.331492i \(-0.107552\pi\)
\(570\) 0 0
\(571\) 9849.25 9849.25i 0.721853 0.721853i −0.247129 0.968983i \(-0.579487\pi\)
0.968983 + 0.247129i \(0.0794872\pi\)
\(572\) 0 0
\(573\) 7112.89 + 7112.89i 0.518578 + 0.518578i
\(574\) 0 0
\(575\) 8726.79 0.632926
\(576\) 0 0
\(577\) −20584.4 −1.48516 −0.742580 0.669757i \(-0.766398\pi\)
−0.742580 + 0.669757i \(0.766398\pi\)
\(578\) 0 0
\(579\) −1038.59 1038.59i −0.0745467 0.0745467i
\(580\) 0 0
\(581\) 27544.1 27544.1i 1.96682 1.96682i
\(582\) 0 0
\(583\) 9439.66i 0.670585i
\(584\) 0 0
\(585\) 1109.08i 0.0783840i
\(586\) 0 0
\(587\) 2586.77 2586.77i 0.181887 0.181887i −0.610291 0.792177i \(-0.708947\pi\)
0.792177 + 0.610291i \(0.208947\pi\)
\(588\) 0 0
\(589\) 486.396 + 486.396i 0.0340265 + 0.0340265i
\(590\) 0 0
\(591\) 4465.54 0.310809
\(592\) 0 0
\(593\) −6035.89 −0.417984 −0.208992 0.977917i \(-0.567018\pi\)
−0.208992 + 0.977917i \(0.567018\pi\)
\(594\) 0 0
\(595\) 1505.28 + 1505.28i 0.103715 + 0.103715i
\(596\) 0 0
\(597\) 4586.95 4586.95i 0.314458 0.314458i
\(598\) 0 0
\(599\) 5427.20i 0.370199i −0.982720 0.185100i \(-0.940739\pi\)
0.982720 0.185100i \(-0.0592608\pi\)
\(600\) 0 0
\(601\) 17725.7i 1.20307i 0.798847 + 0.601535i \(0.205444\pi\)
−0.798847 + 0.601535i \(0.794556\pi\)
\(602\) 0 0
\(603\) −10814.5 + 10814.5i −0.730349 + 0.730349i
\(604\) 0 0
\(605\) −618.793 618.793i −0.0415827 0.0415827i
\(606\) 0 0
\(607\) −13487.6 −0.901884 −0.450942 0.892553i \(-0.648912\pi\)
−0.450942 + 0.892553i \(0.648912\pi\)
\(608\) 0 0
\(609\) −7315.03 −0.486732
\(610\) 0 0
\(611\) −4855.57 4855.57i −0.321498 0.321498i
\(612\) 0 0
\(613\) −16850.4 + 16850.4i −1.11025 + 1.11025i −0.117129 + 0.993117i \(0.537369\pi\)
−0.993117 + 0.117129i \(0.962631\pi\)
\(614\) 0 0
\(615\) 362.918i 0.0237956i
\(616\) 0 0
\(617\) 535.243i 0.0349239i −0.999848 0.0174620i \(-0.994441\pi\)
0.999848 0.0174620i \(-0.00555860\pi\)
\(618\) 0 0
\(619\) 19691.0 19691.0i 1.27859 1.27859i 0.337130 0.941458i \(-0.390544\pi\)
0.941458 0.337130i \(-0.109456\pi\)
\(620\) 0 0
\(621\) 6424.88 + 6424.88i 0.415172 + 0.415172i
\(622\) 0 0
\(623\) −137.538 −0.00884485
\(624\) 0 0
\(625\) −15358.3 −0.982934
\(626\) 0 0
\(627\) 2638.98 + 2638.98i 0.168087 + 0.168087i
\(628\) 0 0
\(629\) −1883.06 + 1883.06i −0.119368 + 0.119368i
\(630\) 0 0
\(631\) 11880.2i 0.749511i 0.927124 + 0.374755i \(0.122273\pi\)
−0.927124 + 0.374755i \(0.877727\pi\)
\(632\) 0 0
\(633\) 5627.72i 0.353368i
\(634\) 0 0
\(635\) 590.132 590.132i 0.0368798 0.0368798i
\(636\) 0 0
\(637\) 24419.5 + 24419.5i 1.51889 + 1.51889i
\(638\) 0 0
\(639\) 8025.45 0.496842
\(640\) 0 0
\(641\) 18341.0 1.13015 0.565074 0.825040i \(-0.308848\pi\)
0.565074 + 0.825040i \(0.308848\pi\)
\(642\) 0 0
\(643\) −7026.21 7026.21i −0.430928 0.430928i 0.458016 0.888944i \(-0.348560\pi\)
−0.888944 + 0.458016i \(0.848560\pi\)
\(644\) 0 0
\(645\) 286.383 286.383i 0.0174827 0.0174827i
\(646\) 0 0
\(647\) 21429.7i 1.30215i 0.759015 + 0.651073i \(0.225681\pi\)
−0.759015 + 0.651073i \(0.774319\pi\)
\(648\) 0 0
\(649\) 7879.56i 0.476579i
\(650\) 0 0
\(651\) −511.613 + 511.613i −0.0308014 + 0.0308014i
\(652\) 0 0
\(653\) 8681.22 + 8681.22i 0.520248 + 0.520248i 0.917646 0.397398i \(-0.130087\pi\)
−0.397398 + 0.917646i \(0.630087\pi\)
\(654\) 0 0
\(655\) 947.284 0.0565091
\(656\) 0 0
\(657\) 7170.92 0.425821
\(658\) 0 0
\(659\) 4151.60 + 4151.60i 0.245407 + 0.245407i 0.819083 0.573675i \(-0.194483\pi\)
−0.573675 + 0.819083i \(0.694483\pi\)
\(660\) 0 0
\(661\) 12239.0 12239.0i 0.720182 0.720182i −0.248460 0.968642i \(-0.579924\pi\)
0.968642 + 0.248460i \(0.0799244\pi\)
\(662\) 0 0
\(663\) 16707.9i 0.978706i
\(664\) 0 0
\(665\) 1903.98i 0.111027i
\(666\) 0 0
\(667\) 4451.75 4451.75i 0.258429 0.258429i
\(668\) 0 0
\(669\) −8559.23 8559.23i −0.494647 0.494647i
\(670\) 0 0
\(671\) 8.22827 0.000473396
\(672\) 0 0
\(673\) 6528.62 0.373937 0.186969 0.982366i \(-0.440134\pi\)
0.186969 + 0.982366i \(0.440134\pi\)
\(674\) 0 0
\(675\) −11372.8 11372.8i −0.648500 0.648500i
\(676\) 0 0
\(677\) −14220.6 + 14220.6i −0.807299 + 0.807299i −0.984224 0.176925i \(-0.943385\pi\)
0.176925 + 0.984224i \(0.443385\pi\)
\(678\) 0 0
\(679\) 11038.2i 0.623867i
\(680\) 0 0
\(681\) 2781.76i 0.156530i
\(682\) 0 0
\(683\) −21419.5 + 21419.5i −1.19999 + 1.19999i −0.225827 + 0.974167i \(0.572508\pi\)
−0.974167 + 0.225827i \(0.927492\pi\)
\(684\) 0 0
\(685\) 952.149 + 952.149i 0.0531091 + 0.0531091i
\(686\) 0 0
\(687\) 2631.60 0.146145
\(688\) 0 0
\(689\) −37797.0 −2.08991
\(690\) 0 0
\(691\) 16537.1 + 16537.1i 0.910423 + 0.910423i 0.996305 0.0858818i \(-0.0273708\pi\)
−0.0858818 + 0.996305i \(0.527371\pi\)
\(692\) 0 0
\(693\) 6748.45 6748.45i 0.369917 0.369917i
\(694\) 0 0
\(695\) 331.484i 0.0180920i
\(696\) 0 0
\(697\) 13291.8i 0.722331i
\(698\) 0 0
\(699\) −6922.92 + 6922.92i −0.374605 + 0.374605i
\(700\) 0 0
\(701\) −3026.52 3026.52i −0.163067 0.163067i 0.620857 0.783924i \(-0.286785\pi\)
−0.783924 + 0.620857i \(0.786785\pi\)
\(702\) 0 0
\(703\) −2381.82 −0.127784
\(704\) 0 0
\(705\) 236.728 0.0126464
\(706\) 0 0
\(707\) 11371.5 + 11371.5i 0.604908 + 0.604908i
\(708\) 0 0
\(709\) 3981.14 3981.14i 0.210881 0.210881i −0.593761 0.804642i \(-0.702357\pi\)
0.804642 + 0.593761i \(0.202357\pi\)
\(710\) 0 0
\(711\) 13501.7i 0.712170i
\(712\) 0 0
\(713\) 622.710i 0.0327078i
\(714\) 0 0
\(715\) −703.160 + 703.160i −0.0367786 + 0.0367786i
\(716\) 0 0
\(717\) 5852.36 + 5852.36i 0.304826 + 0.304826i
\(718\) 0 0
\(719\) 5682.25 0.294732 0.147366 0.989082i \(-0.452921\pi\)
0.147366 + 0.989082i \(0.452921\pi\)
\(720\) 0 0
\(721\) 8955.64 0.462587
\(722\) 0 0
\(723\) 2239.37 + 2239.37i 0.115191 + 0.115191i
\(724\) 0 0
\(725\) −7880.09 + 7880.09i −0.403668 + 0.403668i
\(726\) 0 0
\(727\) 18883.0i 0.963317i −0.876359 0.481658i \(-0.840035\pi\)
0.876359 0.481658i \(-0.159965\pi\)
\(728\) 0 0
\(729\) 6863.99i 0.348727i
\(730\) 0 0
\(731\) 10488.8 10488.8i 0.530698 0.530698i
\(732\) 0 0
\(733\) −24962.5 24962.5i −1.25786 1.25786i −0.952113 0.305748i \(-0.901094\pi\)
−0.305748 0.952113i \(-0.598906\pi\)
\(734\) 0 0
\(735\) −1190.54 −0.0597467
\(736\) 0 0
\(737\) 13712.9 0.685375
\(738\) 0 0
\(739\) −6202.38 6202.38i −0.308739 0.308739i 0.535681 0.844420i \(-0.320055\pi\)
−0.844420 + 0.535681i \(0.820055\pi\)
\(740\) 0 0
\(741\) −10566.6 + 10566.6i −0.523854 + 0.523854i
\(742\) 0 0
\(743\) 30.9140i 0.00152641i 1.00000 0.000763205i \(0.000242936\pi\)
−1.00000 0.000763205i \(0.999757\pi\)
\(744\) 0 0
\(745\) 708.187i 0.0348268i
\(746\) 0 0
\(747\) 18118.8 18118.8i 0.887459 0.887459i
\(748\) 0 0
\(749\) −17496.5 17496.5i −0.853547 0.853547i
\(750\) 0 0
\(751\) 16318.5 0.792905 0.396453 0.918055i \(-0.370241\pi\)
0.396453 + 0.918055i \(0.370241\pi\)
\(752\) 0 0
\(753\) 18335.5 0.887361
\(754\) 0 0
\(755\) −95.8865 95.8865i −0.00462208 0.00462208i
\(756\) 0 0
\(757\) 9854.59 9854.59i 0.473146 0.473146i −0.429786 0.902931i \(-0.641411\pi\)
0.902931 + 0.429786i \(0.141411\pi\)
\(758\) 0 0
\(759\) 3378.57i 0.161573i
\(760\) 0 0
\(761\) 3823.42i 0.182127i 0.995845 + 0.0910637i \(0.0290267\pi\)
−0.995845 + 0.0910637i \(0.970973\pi\)
\(762\) 0 0
\(763\) −27596.4 + 27596.4i −1.30938 + 1.30938i
\(764\) 0 0
\(765\) 990.185 + 990.185i 0.0467977 + 0.0467977i
\(766\) 0 0
\(767\) −31550.2 −1.48528
\(768\) 0 0
\(769\) 31689.1 1.48601 0.743003 0.669288i \(-0.233401\pi\)
0.743003 + 0.669288i \(0.233401\pi\)
\(770\) 0 0
\(771\) −1211.91 1211.91i −0.0566094 0.0566094i
\(772\) 0 0
\(773\) −1305.84 + 1305.84i −0.0607604 + 0.0607604i −0.736834 0.676074i \(-0.763680\pi\)
0.676074 + 0.736834i \(0.263680\pi\)
\(774\) 0 0
\(775\) 1102.27i 0.0510898i
\(776\) 0 0
\(777\) 2505.31i 0.115672i
\(778\) 0 0
\(779\) 8406.21 8406.21i 0.386629 0.386629i
\(780\) 0 0
\(781\) −5088.18 5088.18i −0.233123 0.233123i
\(782\) 0 0
\(783\) −11603.0 −0.529577
\(784\) 0 0
\(785\) 842.465 0.0383043
\(786\) 0 0
\(787\) −14399.5 14399.5i −0.652209 0.652209i 0.301316 0.953524i \(-0.402574\pi\)
−0.953524 + 0.301316i \(0.902574\pi\)
\(788\) 0 0
\(789\) −9865.37 + 9865.37i −0.445141 + 0.445141i
\(790\) 0 0
\(791\) 53047.6i 2.38452i
\(792\) 0 0
\(793\) 32.9465i 0.00147537i
\(794\) 0 0
\(795\) 921.374 921.374i 0.0411041 0.0411041i
\(796\) 0 0
\(797\) −5572.19 5572.19i −0.247650 0.247650i 0.572356 0.820006i \(-0.306030\pi\)
−0.820006 + 0.572356i \(0.806030\pi\)
\(798\) 0 0
\(799\) 8670.13 0.383889
\(800\) 0 0
\(801\) −90.4737 −0.00399093
\(802\) 0 0
\(803\) −4546.40 4546.40i −0.199800 0.199800i
\(804\) 0 0
\(805\) 1218.79 1218.79i 0.0533622 0.0533622i
\(806\) 0 0
\(807\) 3722.81i 0.162390i
\(808\) 0 0
\(809\) 5081.28i 0.220826i 0.993886 + 0.110413i \(0.0352174\pi\)
−0.993886 + 0.110413i \(0.964783\pi\)
\(810\) 0 0
\(811\) −4493.19 + 4493.19i −0.194546 + 0.194546i −0.797657 0.603111i \(-0.793928\pi\)
0.603111 + 0.797657i \(0.293928\pi\)
\(812\) 0 0
\(813\) −7955.35 7955.35i −0.343181 0.343181i
\(814\) 0 0
\(815\) 2235.36 0.0960752
\(816\) 0 0
\(817\) 13266.9 0.568114
\(818\) 0 0
\(819\) 27021.2 + 27021.2i 1.15287 + 1.15287i
\(820\) 0 0
\(821\) 2115.54 2115.54i 0.0899304 0.0899304i −0.660710 0.750641i \(-0.729745\pi\)
0.750641 + 0.660710i \(0.229745\pi\)
\(822\) 0 0
\(823\) 24432.6i 1.03483i −0.855734 0.517416i \(-0.826894\pi\)
0.855734 0.517416i \(-0.173106\pi\)
\(824\) 0 0
\(825\) 5980.44i 0.252378i
\(826\) 0 0
\(827\) −21051.4 + 21051.4i −0.885162 + 0.885162i −0.994054 0.108892i \(-0.965270\pi\)
0.108892 + 0.994054i \(0.465270\pi\)
\(828\) 0 0
\(829\) 25442.0 + 25442.0i 1.06591 + 1.06591i 0.997669 + 0.0682392i \(0.0217381\pi\)
0.0682392 + 0.997669i \(0.478262\pi\)
\(830\) 0 0
\(831\) 14023.7 0.585413
\(832\) 0 0
\(833\) −43603.5 −1.81365
\(834\) 0 0
\(835\) −2299.17 2299.17i −0.0952889 0.0952889i
\(836\) 0 0
\(837\) −811.516 + 811.516i −0.0335127 + 0.0335127i
\(838\) 0 0
\(839\) 18757.2i 0.771837i 0.922533 + 0.385919i \(0.126115\pi\)
−0.922533 + 0.385919i \(0.873885\pi\)
\(840\) 0 0
\(841\) 16349.4i 0.670358i
\(842\) 0 0
\(843\) 14813.9 14813.9i 0.605242 0.605242i
\(844\) 0 0
\(845\) −1504.22 1504.22i −0.0612389 0.0612389i
\(846\) 0 0
\(847\) 30152.2 1.22319
\(848\) 0 0
\(849\) −8922.94 −0.360700
\(850\) 0 0
\(851\) 1524.67 + 1524.67i 0.0614160 + 0.0614160i
\(852\) 0 0
\(853\) −6100.17 + 6100.17i −0.244860 + 0.244860i −0.818857 0.573997i \(-0.805392\pi\)
0.573997 + 0.818857i \(0.305392\pi\)
\(854\) 0 0
\(855\) 1252.45i 0.0500971i
\(856\) 0 0
\(857\) 2079.04i 0.0828687i 0.999141 + 0.0414344i \(0.0131927\pi\)
−0.999141 + 0.0414344i \(0.986807\pi\)
\(858\) 0 0
\(859\) −8301.95 + 8301.95i −0.329754 + 0.329754i −0.852493 0.522739i \(-0.824910\pi\)
0.522739 + 0.852493i \(0.324910\pi\)
\(860\) 0 0
\(861\) 8842.03 + 8842.03i 0.349983 + 0.349983i
\(862\) 0 0
\(863\) −43830.8 −1.72887 −0.864436 0.502743i \(-0.832324\pi\)
−0.864436 + 0.502743i \(0.832324\pi\)
\(864\) 0 0
\(865\) −3134.18 −0.123197
\(866\) 0 0
\(867\) −5171.62 5171.62i −0.202581 0.202581i
\(868\) 0 0
\(869\) −8560.14 + 8560.14i −0.334158 + 0.334158i
\(870\) 0 0
\(871\) 54907.3i 2.13601i
\(872\) 0 0
\(873\) 7261.01i 0.281498i
\(874\) 0 0
\(875\) −4327.14 + 4327.14i −0.167182 + 0.167182i
\(876\) 0 0
\(877\) −2565.89 2565.89i −0.0987958 0.0987958i 0.655981 0.754777i \(-0.272255\pi\)
−0.754777 + 0.655981i \(0.772255\pi\)
\(878\) 0 0
\(879\) 15681.2 0.601723
\(880\) 0 0
\(881\) 26429.8 1.01072 0.505360 0.862909i \(-0.331360\pi\)
0.505360 + 0.862909i \(0.331360\pi\)
\(882\) 0 0
\(883\) 20708.4 + 20708.4i 0.789232 + 0.789232i 0.981368 0.192136i \(-0.0615415\pi\)
−0.192136 + 0.981368i \(0.561542\pi\)
\(884\) 0 0
\(885\) 769.097 769.097i 0.0292123 0.0292123i
\(886\) 0 0
\(887\) 22350.0i 0.846042i 0.906120 + 0.423021i \(0.139030\pi\)
−0.906120 + 0.423021i \(0.860970\pi\)
\(888\) 0 0
\(889\) 28755.6i 1.08485i
\(890\) 0 0
\(891\) 1862.19 1862.19i 0.0700175 0.0700175i
\(892\) 0 0
\(893\) 5483.28 + 5483.28i 0.205477 + 0.205477i
\(894\) 0 0
\(895\) −1476.28 −0.0551360
\(896\) 0 0
\(897\) 13528.0 0.503553
\(898\) 0 0
\(899\) 562.292 + 562.292i 0.0208604 + 0.0208604i
\(900\) 0 0
\(901\) 33745.2 33745.2i 1.24774 1.24774i
\(902\) 0 0
\(903\) 13954.7i 0.514267i
\(904\) 0 0
\(905\) 1879.95i 0.0690516i
\(906\) 0 0
\(907\) 4086.93 4086.93i 0.149619 0.149619i −0.628329 0.777948i \(-0.716261\pi\)
0.777948 + 0.628329i \(0.216261\pi\)
\(908\) 0 0
\(909\) 7480.29 + 7480.29i 0.272944 + 0.272944i
\(910\) 0 0
\(911\) 20024.6 0.728259 0.364130 0.931348i \(-0.381367\pi\)
0.364130 + 0.931348i \(0.381367\pi\)
\(912\) 0 0
\(913\) −22974.8 −0.832810
\(914\) 0 0
\(915\) −0.803134 0.803134i −2.90173e−5 2.90173e-5i
\(916\) 0 0
\(917\) −23079.3 + 23079.3i −0.831131 + 0.831131i
\(918\) 0 0
\(919\) 26977.7i 0.968349i 0.874971 + 0.484175i \(0.160880\pi\)
−0.874971 + 0.484175i \(0.839120\pi\)
\(920\) 0 0
\(921\) 15294.2i 0.547189i
\(922\) 0 0
\(923\) 20373.4 20373.4i 0.726542 0.726542i
\(924\) 0 0
\(925\) −2698.84 2698.84i −0.0959321 0.0959321i
\(926\) 0 0
\(927\) 5891.10 0.208726
\(928\) 0 0
\(929\) −34371.4 −1.21387 −0.606937 0.794750i \(-0.707602\pi\)
−0.606937 + 0.794750i \(0.707602\pi\)
\(930\) 0 0
\(931\) −27576.3 27576.3i −0.970760 0.970760i
\(932\) 0 0
\(933\) 10303.4 10303.4i 0.361541 0.361541i
\(934\) 0 0
\(935\) 1255.57i 0.0439159i
\(936\) 0 0
\(937\) 32901.8i 1.14712i 0.819163 + 0.573561i \(0.194439\pi\)
−0.819163 + 0.573561i \(0.805561\pi\)
\(938\) 0 0
\(939\) 9342.60 9342.60i 0.324690 0.324690i
\(940\) 0 0
\(941\) 33381.6 + 33381.6i 1.15644 + 1.15644i 0.985236 + 0.171204i \(0.0547656\pi\)
0.171204 + 0.985236i \(0.445234\pi\)
\(942\) 0 0
\(943\) −10762.1 −0.371646
\(944\) 0 0
\(945\) −3176.65 −0.109351
\(946\) 0 0
\(947\) −13611.9 13611.9i −0.467084 0.467084i 0.433884 0.900969i \(-0.357143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(948\) 0 0
\(949\) 18204.1 18204.1i 0.622686 0.622686i
\(950\) 0 0
\(951\) 22534.1i 0.768369i
\(952\) 0 0
\(953\) 4572.49i 0.155422i 0.996976 + 0.0777112i \(0.0247612\pi\)
−0.996976 + 0.0777112i \(0.975239\pi\)
\(954\) 0 0
\(955\) 2140.25 2140.25i 0.0725202 0.0725202i
\(956\) 0 0
\(957\) 3050.77 + 3050.77i 0.103048 + 0.103048i
\(958\) 0 0
\(959\) −46395.7 −1.56225
\(960\) 0 0
\(961\) −29712.3 −0.997360
\(962\) 0 0
\(963\) −11509.3 11509.3i −0.385133 0.385133i
\(964\) 0 0
\(965\) −312.510 + 312.510i −0.0104249 + 0.0104249i
\(966\) 0 0
\(967\) 33060.9i 1.09945i −0.835346 0.549725i \(-0.814732\pi\)
0.835346 0.549725i \(-0.185268\pi\)
\(968\) 0 0
\(969\) 18867.9i 0.625514i
\(970\) 0 0
\(971\) 15553.4 15553.4i 0.514041 0.514041i −0.401721 0.915762i \(-0.631588\pi\)
0.915762 + 0.401721i \(0.131588\pi\)
\(972\) 0 0
\(973\) −8076.18 8076.18i −0.266095 0.266095i
\(974\) 0 0
\(975\) −23946.1 −0.786551
\(976\) 0 0
\(977\) 1241.54 0.0406555 0.0203277 0.999793i \(-0.493529\pi\)
0.0203277 + 0.999793i \(0.493529\pi\)
\(978\) 0 0
\(979\) 57.3608 + 57.3608i 0.00187258 + 0.00187258i
\(980\) 0 0
\(981\) −18153.2 + 18153.2i −0.590812 + 0.590812i
\(982\) 0 0
\(983\) 10797.7i 0.350349i 0.984537 + 0.175175i \(0.0560490\pi\)
−0.984537 + 0.175175i \(0.943951\pi\)
\(984\) 0 0
\(985\) 1343.67i 0.0434648i
\(986\) 0 0
\(987\) −5767.56 + 5767.56i −0.186001 + 0.186001i
\(988\) 0 0
\(989\) −8492.49 8492.49i −0.273049 0.273049i
\(990\) 0 0
\(991\) 10204.8 0.327112 0.163556 0.986534i \(-0.447704\pi\)
0.163556 + 0.986534i \(0.447704\pi\)
\(992\) 0 0
\(993\) 7203.49 0.230207
\(994\) 0 0
\(995\) −1380.20 1380.20i −0.0439752 0.0439752i
\(996\) 0 0
\(997\) 15046.7 15046.7i 0.477967 0.477967i −0.426514 0.904481i \(-0.640259\pi\)
0.904481 + 0.426514i \(0.140259\pi\)
\(998\) 0 0
\(999\) 3973.90i 0.125855i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 64.4.e.a.17.2 10
3.2 odd 2 576.4.k.a.145.3 10
4.3 odd 2 16.4.e.a.13.4 yes 10
8.3 odd 2 128.4.e.b.33.2 10
8.5 even 2 128.4.e.a.33.4 10
12.11 even 2 144.4.k.a.109.2 10
16.3 odd 4 128.4.e.b.97.2 10
16.5 even 4 inner 64.4.e.a.49.2 10
16.11 odd 4 16.4.e.a.5.4 10
16.13 even 4 128.4.e.a.97.4 10
32.3 odd 8 1024.4.b.j.513.7 10
32.5 even 8 1024.4.a.m.1.7 10
32.11 odd 8 1024.4.a.n.1.7 10
32.13 even 8 1024.4.b.k.513.7 10
32.19 odd 8 1024.4.b.j.513.4 10
32.21 even 8 1024.4.a.m.1.4 10
32.27 odd 8 1024.4.a.n.1.4 10
32.29 even 8 1024.4.b.k.513.4 10
48.5 odd 4 576.4.k.a.433.3 10
48.11 even 4 144.4.k.a.37.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.4.e.a.5.4 10 16.11 odd 4
16.4.e.a.13.4 yes 10 4.3 odd 2
64.4.e.a.17.2 10 1.1 even 1 trivial
64.4.e.a.49.2 10 16.5 even 4 inner
128.4.e.a.33.4 10 8.5 even 2
128.4.e.a.97.4 10 16.13 even 4
128.4.e.b.33.2 10 8.3 odd 2
128.4.e.b.97.2 10 16.3 odd 4
144.4.k.a.37.2 10 48.11 even 4
144.4.k.a.109.2 10 12.11 even 2
576.4.k.a.145.3 10 3.2 odd 2
576.4.k.a.433.3 10 48.5 odd 4
1024.4.a.m.1.4 10 32.21 even 8
1024.4.a.m.1.7 10 32.5 even 8
1024.4.a.n.1.4 10 32.27 odd 8
1024.4.a.n.1.7 10 32.11 odd 8
1024.4.b.j.513.4 10 32.19 odd 8
1024.4.b.j.513.7 10 32.3 odd 8
1024.4.b.k.513.4 10 32.29 even 8
1024.4.b.k.513.7 10 32.13 even 8