Properties

Label 1014.4.b.c.337.2
Level $1014$
Weight $4$
Character 1014.337
Analytic conductor $59.828$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1014,4,Mod(337,1014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1014, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1014.337");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1014.b (of order \(2\), degree \(1\), 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: \(59.8279367458\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1014.337
Dual form 1014.4.b.c.337.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000i q^{2} -3.00000 q^{3} -4.00000 q^{4} +6.00000i q^{5} -6.00000i q^{6} -20.0000i q^{7} -8.00000i q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.00000i q^{2} -3.00000 q^{3} -4.00000 q^{4} +6.00000i q^{5} -6.00000i q^{6} -20.0000i q^{7} -8.00000i q^{8} +9.00000 q^{9} -12.0000 q^{10} -24.0000i q^{11} +12.0000 q^{12} +40.0000 q^{14} -18.0000i q^{15} +16.0000 q^{16} +30.0000 q^{17} +18.0000i q^{18} -16.0000i q^{19} -24.0000i q^{20} +60.0000i q^{21} +48.0000 q^{22} +72.0000 q^{23} +24.0000i q^{24} +89.0000 q^{25} -27.0000 q^{27} +80.0000i q^{28} -282.000 q^{29} +36.0000 q^{30} +164.000i q^{31} +32.0000i q^{32} +72.0000i q^{33} +60.0000i q^{34} +120.000 q^{35} -36.0000 q^{36} -110.000i q^{37} +32.0000 q^{38} +48.0000 q^{40} -126.000i q^{41} -120.000 q^{42} -164.000 q^{43} +96.0000i q^{44} +54.0000i q^{45} +144.000i q^{46} +204.000i q^{47} -48.0000 q^{48} -57.0000 q^{49} +178.000i q^{50} -90.0000 q^{51} -738.000 q^{53} -54.0000i q^{54} +144.000 q^{55} -160.000 q^{56} +48.0000i q^{57} -564.000i q^{58} -120.000i q^{59} +72.0000i q^{60} +614.000 q^{61} -328.000 q^{62} -180.000i q^{63} -64.0000 q^{64} -144.000 q^{66} +848.000i q^{67} -120.000 q^{68} -216.000 q^{69} +240.000i q^{70} +132.000i q^{71} -72.0000i q^{72} -218.000i q^{73} +220.000 q^{74} -267.000 q^{75} +64.0000i q^{76} -480.000 q^{77} -1096.00 q^{79} +96.0000i q^{80} +81.0000 q^{81} +252.000 q^{82} +552.000i q^{83} -240.000i q^{84} +180.000i q^{85} -328.000i q^{86} +846.000 q^{87} -192.000 q^{88} -210.000i q^{89} -108.000 q^{90} -288.000 q^{92} -492.000i q^{93} -408.000 q^{94} +96.0000 q^{95} -96.0000i q^{96} -1726.00i q^{97} -114.000i q^{98} -216.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 8 q^{4} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 8 q^{4} + 18 q^{9} - 24 q^{10} + 24 q^{12} + 80 q^{14} + 32 q^{16} + 60 q^{17} + 96 q^{22} + 144 q^{23} + 178 q^{25} - 54 q^{27} - 564 q^{29} + 72 q^{30} + 240 q^{35} - 72 q^{36} + 64 q^{38} + 96 q^{40} - 240 q^{42} - 328 q^{43} - 96 q^{48} - 114 q^{49} - 180 q^{51} - 1476 q^{53} + 288 q^{55} - 320 q^{56} + 1228 q^{61} - 656 q^{62} - 128 q^{64} - 288 q^{66} - 240 q^{68} - 432 q^{69} + 440 q^{74} - 534 q^{75} - 960 q^{77} - 2192 q^{79} + 162 q^{81} + 504 q^{82} + 1692 q^{87} - 384 q^{88} - 216 q^{90} - 576 q^{92} - 816 q^{94} + 192 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 0.707107i
\(3\) −3.00000 −0.577350
\(4\) −4.00000 −0.500000
\(5\) 6.00000i 0.536656i 0.963328 + 0.268328i \(0.0864711\pi\)
−0.963328 + 0.268328i \(0.913529\pi\)
\(6\) − 6.00000i − 0.408248i
\(7\) − 20.0000i − 1.07990i −0.841698 0.539949i \(-0.818443\pi\)
0.841698 0.539949i \(-0.181557\pi\)
\(8\) − 8.00000i − 0.353553i
\(9\) 9.00000 0.333333
\(10\) −12.0000 −0.379473
\(11\) − 24.0000i − 0.657843i −0.944357 0.328921i \(-0.893315\pi\)
0.944357 0.328921i \(-0.106685\pi\)
\(12\) 12.0000 0.288675
\(13\) 0 0
\(14\) 40.0000 0.763604
\(15\) − 18.0000i − 0.309839i
\(16\) 16.0000 0.250000
\(17\) 30.0000 0.428004 0.214002 0.976833i \(-0.431350\pi\)
0.214002 + 0.976833i \(0.431350\pi\)
\(18\) 18.0000i 0.235702i
\(19\) − 16.0000i − 0.193192i −0.995324 0.0965961i \(-0.969204\pi\)
0.995324 0.0965961i \(-0.0307955\pi\)
\(20\) − 24.0000i − 0.268328i
\(21\) 60.0000i 0.623480i
\(22\) 48.0000 0.465165
\(23\) 72.0000 0.652741 0.326370 0.945242i \(-0.394174\pi\)
0.326370 + 0.945242i \(0.394174\pi\)
\(24\) 24.0000i 0.204124i
\(25\) 89.0000 0.712000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 80.0000i 0.539949i
\(29\) −282.000 −1.80573 −0.902864 0.429927i \(-0.858539\pi\)
−0.902864 + 0.429927i \(0.858539\pi\)
\(30\) 36.0000 0.219089
\(31\) 164.000i 0.950170i 0.879940 + 0.475085i \(0.157583\pi\)
−0.879940 + 0.475085i \(0.842417\pi\)
\(32\) 32.0000i 0.176777i
\(33\) 72.0000i 0.379806i
\(34\) 60.0000i 0.302645i
\(35\) 120.000 0.579534
\(36\) −36.0000 −0.166667
\(37\) − 110.000i − 0.488754i −0.969680 0.244377i \(-0.921417\pi\)
0.969680 0.244377i \(-0.0785834\pi\)
\(38\) 32.0000 0.136608
\(39\) 0 0
\(40\) 48.0000 0.189737
\(41\) − 126.000i − 0.479949i −0.970779 0.239974i \(-0.922861\pi\)
0.970779 0.239974i \(-0.0771390\pi\)
\(42\) −120.000 −0.440867
\(43\) −164.000 −0.581622 −0.290811 0.956780i \(-0.593925\pi\)
−0.290811 + 0.956780i \(0.593925\pi\)
\(44\) 96.0000i 0.328921i
\(45\) 54.0000i 0.178885i
\(46\) 144.000i 0.461557i
\(47\) 204.000i 0.633116i 0.948573 + 0.316558i \(0.102527\pi\)
−0.948573 + 0.316558i \(0.897473\pi\)
\(48\) −48.0000 −0.144338
\(49\) −57.0000 −0.166181
\(50\) 178.000i 0.503460i
\(51\) −90.0000 −0.247108
\(52\) 0 0
\(53\) −738.000 −1.91268 −0.956341 0.292255i \(-0.905595\pi\)
−0.956341 + 0.292255i \(0.905595\pi\)
\(54\) − 54.0000i − 0.136083i
\(55\) 144.000 0.353036
\(56\) −160.000 −0.381802
\(57\) 48.0000i 0.111540i
\(58\) − 564.000i − 1.27684i
\(59\) − 120.000i − 0.264791i −0.991197 0.132396i \(-0.957733\pi\)
0.991197 0.132396i \(-0.0422669\pi\)
\(60\) 72.0000i 0.154919i
\(61\) 614.000 1.28876 0.644382 0.764703i \(-0.277115\pi\)
0.644382 + 0.764703i \(0.277115\pi\)
\(62\) −328.000 −0.671872
\(63\) − 180.000i − 0.359966i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) −144.000 −0.268563
\(67\) 848.000i 1.54626i 0.634245 + 0.773132i \(0.281311\pi\)
−0.634245 + 0.773132i \(0.718689\pi\)
\(68\) −120.000 −0.214002
\(69\) −216.000 −0.376860
\(70\) 240.000i 0.409793i
\(71\) 132.000i 0.220641i 0.993896 + 0.110321i \(0.0351877\pi\)
−0.993896 + 0.110321i \(0.964812\pi\)
\(72\) − 72.0000i − 0.117851i
\(73\) − 218.000i − 0.349520i −0.984611 0.174760i \(-0.944085\pi\)
0.984611 0.174760i \(-0.0559150\pi\)
\(74\) 220.000 0.345601
\(75\) −267.000 −0.411073
\(76\) 64.0000i 0.0965961i
\(77\) −480.000 −0.710404
\(78\) 0 0
\(79\) −1096.00 −1.56088 −0.780441 0.625230i \(-0.785005\pi\)
−0.780441 + 0.625230i \(0.785005\pi\)
\(80\) 96.0000i 0.134164i
\(81\) 81.0000 0.111111
\(82\) 252.000 0.339375
\(83\) 552.000i 0.729998i 0.931008 + 0.364999i \(0.118931\pi\)
−0.931008 + 0.364999i \(0.881069\pi\)
\(84\) − 240.000i − 0.311740i
\(85\) 180.000i 0.229691i
\(86\) − 328.000i − 0.411269i
\(87\) 846.000 1.04254
\(88\) −192.000 −0.232583
\(89\) − 210.000i − 0.250112i −0.992150 0.125056i \(-0.960089\pi\)
0.992150 0.125056i \(-0.0399110\pi\)
\(90\) −108.000 −0.126491
\(91\) 0 0
\(92\) −288.000 −0.326370
\(93\) − 492.000i − 0.548581i
\(94\) −408.000 −0.447681
\(95\) 96.0000 0.103678
\(96\) − 96.0000i − 0.102062i
\(97\) − 1726.00i − 1.80669i −0.428917 0.903344i \(-0.641105\pi\)
0.428917 0.903344i \(-0.358895\pi\)
\(98\) − 114.000i − 0.117508i
\(99\) − 216.000i − 0.219281i
\(100\) −356.000 −0.356000
\(101\) −798.000 −0.786178 −0.393089 0.919500i \(-0.628594\pi\)
−0.393089 + 0.919500i \(0.628594\pi\)
\(102\) − 180.000i − 0.174732i
\(103\) 520.000 0.497448 0.248724 0.968574i \(-0.419989\pi\)
0.248724 + 0.968574i \(0.419989\pi\)
\(104\) 0 0
\(105\) −360.000 −0.334594
\(106\) − 1476.00i − 1.35247i
\(107\) 12.0000 0.0108419 0.00542095 0.999985i \(-0.498274\pi\)
0.00542095 + 0.999985i \(0.498274\pi\)
\(108\) 108.000 0.0962250
\(109\) − 1834.00i − 1.61161i −0.592182 0.805804i \(-0.701733\pi\)
0.592182 0.805804i \(-0.298267\pi\)
\(110\) 288.000i 0.249634i
\(111\) 330.000i 0.282182i
\(112\) − 320.000i − 0.269975i
\(113\) −366.000 −0.304694 −0.152347 0.988327i \(-0.548683\pi\)
−0.152347 + 0.988327i \(0.548683\pi\)
\(114\) −96.0000 −0.0788704
\(115\) 432.000i 0.350297i
\(116\) 1128.00 0.902864
\(117\) 0 0
\(118\) 240.000 0.187236
\(119\) − 600.000i − 0.462201i
\(120\) −144.000 −0.109545
\(121\) 755.000 0.567243
\(122\) 1228.00i 0.911294i
\(123\) 378.000i 0.277098i
\(124\) − 656.000i − 0.475085i
\(125\) 1284.00i 0.918756i
\(126\) 360.000 0.254535
\(127\) −2144.00 −1.49803 −0.749013 0.662556i \(-0.769472\pi\)
−0.749013 + 0.662556i \(0.769472\pi\)
\(128\) − 128.000i − 0.0883883i
\(129\) 492.000 0.335800
\(130\) 0 0
\(131\) −2748.00 −1.83278 −0.916389 0.400289i \(-0.868910\pi\)
−0.916389 + 0.400289i \(0.868910\pi\)
\(132\) − 288.000i − 0.189903i
\(133\) −320.000 −0.208628
\(134\) −1696.00 −1.09337
\(135\) − 162.000i − 0.103280i
\(136\) − 240.000i − 0.151322i
\(137\) − 2754.00i − 1.71745i −0.512440 0.858723i \(-0.671258\pi\)
0.512440 0.858723i \(-0.328742\pi\)
\(138\) − 432.000i − 0.266480i
\(139\) 2252.00 1.37419 0.687094 0.726568i \(-0.258886\pi\)
0.687094 + 0.726568i \(0.258886\pi\)
\(140\) −480.000 −0.289767
\(141\) − 612.000i − 0.365530i
\(142\) −264.000 −0.156017
\(143\) 0 0
\(144\) 144.000 0.0833333
\(145\) − 1692.00i − 0.969055i
\(146\) 436.000 0.247148
\(147\) 171.000 0.0959445
\(148\) 440.000i 0.244377i
\(149\) − 1770.00i − 0.973182i −0.873630 0.486591i \(-0.838240\pi\)
0.873630 0.486591i \(-0.161760\pi\)
\(150\) − 534.000i − 0.290673i
\(151\) 988.000i 0.532466i 0.963909 + 0.266233i \(0.0857790\pi\)
−0.963909 + 0.266233i \(0.914221\pi\)
\(152\) −128.000 −0.0683038
\(153\) 270.000 0.142668
\(154\) − 960.000i − 0.502331i
\(155\) −984.000 −0.509915
\(156\) 0 0
\(157\) 326.000 0.165717 0.0828587 0.996561i \(-0.473595\pi\)
0.0828587 + 0.996561i \(0.473595\pi\)
\(158\) − 2192.00i − 1.10371i
\(159\) 2214.00 1.10429
\(160\) −192.000 −0.0948683
\(161\) − 1440.00i − 0.704894i
\(162\) 162.000i 0.0785674i
\(163\) − 1496.00i − 0.718870i −0.933170 0.359435i \(-0.882969\pi\)
0.933170 0.359435i \(-0.117031\pi\)
\(164\) 504.000i 0.239974i
\(165\) −432.000 −0.203825
\(166\) −1104.00 −0.516187
\(167\) − 1116.00i − 0.517118i −0.965995 0.258559i \(-0.916752\pi\)
0.965995 0.258559i \(-0.0832476\pi\)
\(168\) 480.000 0.220433
\(169\) 0 0
\(170\) −360.000 −0.162416
\(171\) − 144.000i − 0.0643974i
\(172\) 656.000 0.290811
\(173\) −4374.00 −1.92225 −0.961124 0.276116i \(-0.910953\pi\)
−0.961124 + 0.276116i \(0.910953\pi\)
\(174\) 1692.00i 0.737185i
\(175\) − 1780.00i − 0.768888i
\(176\) − 384.000i − 0.164461i
\(177\) 360.000i 0.152877i
\(178\) 420.000 0.176856
\(179\) −12.0000 −0.00501074 −0.00250537 0.999997i \(-0.500797\pi\)
−0.00250537 + 0.999997i \(0.500797\pi\)
\(180\) − 216.000i − 0.0894427i
\(181\) −4718.00 −1.93749 −0.968746 0.248053i \(-0.920209\pi\)
−0.968746 + 0.248053i \(0.920209\pi\)
\(182\) 0 0
\(183\) −1842.00 −0.744069
\(184\) − 576.000i − 0.230779i
\(185\) 660.000 0.262293
\(186\) 984.000 0.387905
\(187\) − 720.000i − 0.281559i
\(188\) − 816.000i − 0.316558i
\(189\) 540.000i 0.207827i
\(190\) 192.000i 0.0733113i
\(191\) −1368.00 −0.518246 −0.259123 0.965844i \(-0.583434\pi\)
−0.259123 + 0.965844i \(0.583434\pi\)
\(192\) 192.000 0.0721688
\(193\) 3310.00i 1.23450i 0.786766 + 0.617251i \(0.211754\pi\)
−0.786766 + 0.617251i \(0.788246\pi\)
\(194\) 3452.00 1.27752
\(195\) 0 0
\(196\) 228.000 0.0830904
\(197\) 3126.00i 1.13055i 0.824903 + 0.565275i \(0.191230\pi\)
−0.824903 + 0.565275i \(0.808770\pi\)
\(198\) 432.000 0.155055
\(199\) −4664.00 −1.66142 −0.830709 0.556707i \(-0.812065\pi\)
−0.830709 + 0.556707i \(0.812065\pi\)
\(200\) − 712.000i − 0.251730i
\(201\) − 2544.00i − 0.892736i
\(202\) − 1596.00i − 0.555912i
\(203\) 5640.00i 1.95000i
\(204\) 360.000 0.123554
\(205\) 756.000 0.257567
\(206\) 1040.00i 0.351749i
\(207\) 648.000 0.217580
\(208\) 0 0
\(209\) −384.000 −0.127090
\(210\) − 720.000i − 0.236594i
\(211\) −556.000 −0.181406 −0.0907029 0.995878i \(-0.528911\pi\)
−0.0907029 + 0.995878i \(0.528911\pi\)
\(212\) 2952.00 0.956341
\(213\) − 396.000i − 0.127387i
\(214\) 24.0000i 0.00766638i
\(215\) − 984.000i − 0.312131i
\(216\) 216.000i 0.0680414i
\(217\) 3280.00 1.02609
\(218\) 3668.00 1.13958
\(219\) 654.000i 0.201796i
\(220\) −576.000 −0.176518
\(221\) 0 0
\(222\) −660.000 −0.199533
\(223\) − 268.000i − 0.0804781i −0.999190 0.0402390i \(-0.987188\pi\)
0.999190 0.0402390i \(-0.0128119\pi\)
\(224\) 640.000 0.190901
\(225\) 801.000 0.237333
\(226\) − 732.000i − 0.215451i
\(227\) 1800.00i 0.526300i 0.964755 + 0.263150i \(0.0847615\pi\)
−0.964755 + 0.263150i \(0.915239\pi\)
\(228\) − 192.000i − 0.0557698i
\(229\) − 2990.00i − 0.862816i −0.902157 0.431408i \(-0.858017\pi\)
0.902157 0.431408i \(-0.141983\pi\)
\(230\) −864.000 −0.247698
\(231\) 1440.00 0.410152
\(232\) 2256.00i 0.638421i
\(233\) −2826.00 −0.794581 −0.397291 0.917693i \(-0.630049\pi\)
−0.397291 + 0.917693i \(0.630049\pi\)
\(234\) 0 0
\(235\) −1224.00 −0.339766
\(236\) 480.000i 0.132396i
\(237\) 3288.00 0.901175
\(238\) 1200.00 0.326825
\(239\) − 1812.00i − 0.490412i −0.969471 0.245206i \(-0.921144\pi\)
0.969471 0.245206i \(-0.0788556\pi\)
\(240\) − 288.000i − 0.0774597i
\(241\) 1582.00i 0.422845i 0.977395 + 0.211422i \(0.0678096\pi\)
−0.977395 + 0.211422i \(0.932190\pi\)
\(242\) 1510.00i 0.401101i
\(243\) −243.000 −0.0641500
\(244\) −2456.00 −0.644382
\(245\) − 342.000i − 0.0891820i
\(246\) −756.000 −0.195938
\(247\) 0 0
\(248\) 1312.00 0.335936
\(249\) − 1656.00i − 0.421465i
\(250\) −2568.00 −0.649658
\(251\) −2148.00 −0.540162 −0.270081 0.962838i \(-0.587050\pi\)
−0.270081 + 0.962838i \(0.587050\pi\)
\(252\) 720.000i 0.179983i
\(253\) − 1728.00i − 0.429401i
\(254\) − 4288.00i − 1.05926i
\(255\) − 540.000i − 0.132612i
\(256\) 256.000 0.0625000
\(257\) 558.000 0.135436 0.0677181 0.997704i \(-0.478428\pi\)
0.0677181 + 0.997704i \(0.478428\pi\)
\(258\) 984.000i 0.237446i
\(259\) −2200.00 −0.527804
\(260\) 0 0
\(261\) −2538.00 −0.601909
\(262\) − 5496.00i − 1.29597i
\(263\) 2112.00 0.495177 0.247588 0.968865i \(-0.420362\pi\)
0.247588 + 0.968865i \(0.420362\pi\)
\(264\) 576.000 0.134282
\(265\) − 4428.00i − 1.02645i
\(266\) − 640.000i − 0.147522i
\(267\) 630.000i 0.144402i
\(268\) − 3392.00i − 0.773132i
\(269\) 5046.00 1.14372 0.571859 0.820352i \(-0.306223\pi\)
0.571859 + 0.820352i \(0.306223\pi\)
\(270\) 324.000 0.0730297
\(271\) 3796.00i 0.850888i 0.904985 + 0.425444i \(0.139882\pi\)
−0.904985 + 0.425444i \(0.860118\pi\)
\(272\) 480.000 0.107001
\(273\) 0 0
\(274\) 5508.00 1.21442
\(275\) − 2136.00i − 0.468384i
\(276\) 864.000 0.188430
\(277\) −5582.00 −1.21079 −0.605397 0.795924i \(-0.706986\pi\)
−0.605397 + 0.795924i \(0.706986\pi\)
\(278\) 4504.00i 0.971698i
\(279\) 1476.00i 0.316723i
\(280\) − 960.000i − 0.204896i
\(281\) 1950.00i 0.413976i 0.978343 + 0.206988i \(0.0663661\pi\)
−0.978343 + 0.206988i \(0.933634\pi\)
\(282\) 1224.00 0.258469
\(283\) 4732.00 0.993951 0.496976 0.867765i \(-0.334444\pi\)
0.496976 + 0.867765i \(0.334444\pi\)
\(284\) − 528.000i − 0.110321i
\(285\) −288.000 −0.0598584
\(286\) 0 0
\(287\) −2520.00 −0.518296
\(288\) 288.000i 0.0589256i
\(289\) −4013.00 −0.816813
\(290\) 3384.00 0.685225
\(291\) 5178.00i 1.04309i
\(292\) 872.000i 0.174760i
\(293\) − 4998.00i − 0.996540i −0.867022 0.498270i \(-0.833969\pi\)
0.867022 0.498270i \(-0.166031\pi\)
\(294\) 342.000i 0.0678430i
\(295\) 720.000 0.142102
\(296\) −880.000 −0.172801
\(297\) 648.000i 0.126602i
\(298\) 3540.00 0.688143
\(299\) 0 0
\(300\) 1068.00 0.205537
\(301\) 3280.00i 0.628093i
\(302\) −1976.00 −0.376510
\(303\) 2394.00 0.453900
\(304\) − 256.000i − 0.0482980i
\(305\) 3684.00i 0.691624i
\(306\) 540.000i 0.100882i
\(307\) − 6824.00i − 1.26862i −0.773079 0.634310i \(-0.781284\pi\)
0.773079 0.634310i \(-0.218716\pi\)
\(308\) 1920.00 0.355202
\(309\) −1560.00 −0.287202
\(310\) − 1968.00i − 0.360564i
\(311\) 8760.00 1.59722 0.798608 0.601852i \(-0.205570\pi\)
0.798608 + 0.601852i \(0.205570\pi\)
\(312\) 0 0
\(313\) 3962.00 0.715481 0.357740 0.933821i \(-0.383547\pi\)
0.357740 + 0.933821i \(0.383547\pi\)
\(314\) 652.000i 0.117180i
\(315\) 1080.00 0.193178
\(316\) 4384.00 0.780441
\(317\) 7086.00i 1.25549i 0.778420 + 0.627744i \(0.216021\pi\)
−0.778420 + 0.627744i \(0.783979\pi\)
\(318\) 4428.00i 0.780849i
\(319\) 6768.00i 1.18788i
\(320\) − 384.000i − 0.0670820i
\(321\) −36.0000 −0.00625958
\(322\) 2880.00 0.498435
\(323\) − 480.000i − 0.0826870i
\(324\) −324.000 −0.0555556
\(325\) 0 0
\(326\) 2992.00 0.508318
\(327\) 5502.00i 0.930463i
\(328\) −1008.00 −0.169687
\(329\) 4080.00 0.683701
\(330\) − 864.000i − 0.144126i
\(331\) − 9016.00i − 1.49717i −0.663037 0.748586i \(-0.730733\pi\)
0.663037 0.748586i \(-0.269267\pi\)
\(332\) − 2208.00i − 0.364999i
\(333\) − 990.000i − 0.162918i
\(334\) 2232.00 0.365658
\(335\) −5088.00 −0.829812
\(336\) 960.000i 0.155870i
\(337\) −2306.00 −0.372747 −0.186374 0.982479i \(-0.559673\pi\)
−0.186374 + 0.982479i \(0.559673\pi\)
\(338\) 0 0
\(339\) 1098.00 0.175915
\(340\) − 720.000i − 0.114846i
\(341\) 3936.00 0.625063
\(342\) 288.000 0.0455358
\(343\) − 5720.00i − 0.900440i
\(344\) 1312.00i 0.205635i
\(345\) − 1296.00i − 0.202244i
\(346\) − 8748.00i − 1.35924i
\(347\) −11076.0 −1.71352 −0.856759 0.515717i \(-0.827526\pi\)
−0.856759 + 0.515717i \(0.827526\pi\)
\(348\) −3384.00 −0.521269
\(349\) − 2342.00i − 0.359210i −0.983739 0.179605i \(-0.942518\pi\)
0.983739 0.179605i \(-0.0574820\pi\)
\(350\) 3560.00 0.543686
\(351\) 0 0
\(352\) 768.000 0.116291
\(353\) 4650.00i 0.701118i 0.936541 + 0.350559i \(0.114008\pi\)
−0.936541 + 0.350559i \(0.885992\pi\)
\(354\) −720.000 −0.108100
\(355\) −792.000 −0.118408
\(356\) 840.000i 0.125056i
\(357\) 1800.00i 0.266852i
\(358\) − 24.0000i − 0.00354313i
\(359\) 11268.0i 1.65655i 0.560320 + 0.828276i \(0.310678\pi\)
−0.560320 + 0.828276i \(0.689322\pi\)
\(360\) 432.000 0.0632456
\(361\) 6603.00 0.962677
\(362\) − 9436.00i − 1.37001i
\(363\) −2265.00 −0.327498
\(364\) 0 0
\(365\) 1308.00 0.187572
\(366\) − 3684.00i − 0.526136i
\(367\) −7288.00 −1.03660 −0.518298 0.855200i \(-0.673434\pi\)
−0.518298 + 0.855200i \(0.673434\pi\)
\(368\) 1152.00 0.163185
\(369\) − 1134.00i − 0.159983i
\(370\) 1320.00i 0.185469i
\(371\) 14760.0i 2.06550i
\(372\) 1968.00i 0.274290i
\(373\) −9970.00 −1.38399 −0.691993 0.721904i \(-0.743267\pi\)
−0.691993 + 0.721904i \(0.743267\pi\)
\(374\) 1440.00 0.199093
\(375\) − 3852.00i − 0.530444i
\(376\) 1632.00 0.223840
\(377\) 0 0
\(378\) −1080.00 −0.146956
\(379\) 13448.0i 1.82263i 0.411708 + 0.911316i \(0.364932\pi\)
−0.411708 + 0.911316i \(0.635068\pi\)
\(380\) −384.000 −0.0518389
\(381\) 6432.00 0.864885
\(382\) − 2736.00i − 0.366455i
\(383\) 11820.0i 1.57696i 0.615064 + 0.788478i \(0.289130\pi\)
−0.615064 + 0.788478i \(0.710870\pi\)
\(384\) 384.000i 0.0510310i
\(385\) − 2880.00i − 0.381243i
\(386\) −6620.00 −0.872925
\(387\) −1476.00 −0.193874
\(388\) 6904.00i 0.903344i
\(389\) −174.000 −0.0226790 −0.0113395 0.999936i \(-0.503610\pi\)
−0.0113395 + 0.999936i \(0.503610\pi\)
\(390\) 0 0
\(391\) 2160.00 0.279376
\(392\) 456.000i 0.0587538i
\(393\) 8244.00 1.05815
\(394\) −6252.00 −0.799419
\(395\) − 6576.00i − 0.837657i
\(396\) 864.000i 0.109640i
\(397\) 2986.00i 0.377489i 0.982026 + 0.188744i \(0.0604418\pi\)
−0.982026 + 0.188744i \(0.939558\pi\)
\(398\) − 9328.00i − 1.17480i
\(399\) 960.000 0.120451
\(400\) 1424.00 0.178000
\(401\) 10566.0i 1.31581i 0.753100 + 0.657906i \(0.228558\pi\)
−0.753100 + 0.657906i \(0.771442\pi\)
\(402\) 5088.00 0.631260
\(403\) 0 0
\(404\) 3192.00 0.393089
\(405\) 486.000i 0.0596285i
\(406\) −11280.0 −1.37886
\(407\) −2640.00 −0.321523
\(408\) 720.000i 0.0873660i
\(409\) − 7270.00i − 0.878920i −0.898262 0.439460i \(-0.855170\pi\)
0.898262 0.439460i \(-0.144830\pi\)
\(410\) 1512.00i 0.182128i
\(411\) 8262.00i 0.991568i
\(412\) −2080.00 −0.248724
\(413\) −2400.00 −0.285947
\(414\) 1296.00i 0.153852i
\(415\) −3312.00 −0.391758
\(416\) 0 0
\(417\) −6756.00 −0.793388
\(418\) − 768.000i − 0.0898663i
\(419\) −7308.00 −0.852074 −0.426037 0.904706i \(-0.640091\pi\)
−0.426037 + 0.904706i \(0.640091\pi\)
\(420\) 1440.00 0.167297
\(421\) − 5938.00i − 0.687412i −0.939077 0.343706i \(-0.888318\pi\)
0.939077 0.343706i \(-0.111682\pi\)
\(422\) − 1112.00i − 0.128273i
\(423\) 1836.00i 0.211039i
\(424\) 5904.00i 0.676235i
\(425\) 2670.00 0.304739
\(426\) 792.000 0.0900764
\(427\) − 12280.0i − 1.39174i
\(428\) −48.0000 −0.00542095
\(429\) 0 0
\(430\) 1968.00 0.220710
\(431\) 11532.0i 1.28881i 0.764685 + 0.644405i \(0.222895\pi\)
−0.764685 + 0.644405i \(0.777105\pi\)
\(432\) −432.000 −0.0481125
\(433\) 718.000 0.0796879 0.0398440 0.999206i \(-0.487314\pi\)
0.0398440 + 0.999206i \(0.487314\pi\)
\(434\) 6560.00i 0.725553i
\(435\) 5076.00i 0.559484i
\(436\) 7336.00i 0.805804i
\(437\) − 1152.00i − 0.126104i
\(438\) −1308.00 −0.142691
\(439\) −8984.00 −0.976726 −0.488363 0.872640i \(-0.662406\pi\)
−0.488363 + 0.872640i \(0.662406\pi\)
\(440\) − 1152.00i − 0.124817i
\(441\) −513.000 −0.0553936
\(442\) 0 0
\(443\) 2604.00 0.279277 0.139639 0.990203i \(-0.455406\pi\)
0.139639 + 0.990203i \(0.455406\pi\)
\(444\) − 1320.00i − 0.141091i
\(445\) 1260.00 0.134224
\(446\) 536.000 0.0569066
\(447\) 5310.00i 0.561867i
\(448\) 1280.00i 0.134987i
\(449\) 13206.0i 1.38804i 0.719956 + 0.694020i \(0.244162\pi\)
−0.719956 + 0.694020i \(0.755838\pi\)
\(450\) 1602.00i 0.167820i
\(451\) −3024.00 −0.315731
\(452\) 1464.00 0.152347
\(453\) − 2964.00i − 0.307419i
\(454\) −3600.00 −0.372151
\(455\) 0 0
\(456\) 384.000 0.0394352
\(457\) 8426.00i 0.862476i 0.902238 + 0.431238i \(0.141923\pi\)
−0.902238 + 0.431238i \(0.858077\pi\)
\(458\) 5980.00 0.610103
\(459\) −810.000 −0.0823694
\(460\) − 1728.00i − 0.175149i
\(461\) 16686.0i 1.68578i 0.538086 + 0.842890i \(0.319148\pi\)
−0.538086 + 0.842890i \(0.680852\pi\)
\(462\) 2880.00i 0.290021i
\(463\) − 15932.0i − 1.59919i −0.600543 0.799593i \(-0.705049\pi\)
0.600543 0.799593i \(-0.294951\pi\)
\(464\) −4512.00 −0.451432
\(465\) 2952.00 0.294399
\(466\) − 5652.00i − 0.561854i
\(467\) −18540.0 −1.83711 −0.918553 0.395297i \(-0.870642\pi\)
−0.918553 + 0.395297i \(0.870642\pi\)
\(468\) 0 0
\(469\) 16960.0 1.66981
\(470\) − 2448.00i − 0.240251i
\(471\) −978.000 −0.0956770
\(472\) −960.000 −0.0936178
\(473\) 3936.00i 0.382616i
\(474\) 6576.00i 0.637227i
\(475\) − 1424.00i − 0.137553i
\(476\) 2400.00i 0.231100i
\(477\) −6642.00 −0.637560
\(478\) 3624.00 0.346774
\(479\) − 6180.00i − 0.589502i −0.955574 0.294751i \(-0.904763\pi\)
0.955574 0.294751i \(-0.0952367\pi\)
\(480\) 576.000 0.0547723
\(481\) 0 0
\(482\) −3164.00 −0.298996
\(483\) 4320.00i 0.406971i
\(484\) −3020.00 −0.283621
\(485\) 10356.0 0.969571
\(486\) − 486.000i − 0.0453609i
\(487\) 11756.0i 1.09387i 0.837175 + 0.546936i \(0.184206\pi\)
−0.837175 + 0.546936i \(0.815794\pi\)
\(488\) − 4912.00i − 0.455647i
\(489\) 4488.00i 0.415040i
\(490\) 684.000 0.0630612
\(491\) −1908.00 −0.175370 −0.0876852 0.996148i \(-0.527947\pi\)
−0.0876852 + 0.996148i \(0.527947\pi\)
\(492\) − 1512.00i − 0.138549i
\(493\) −8460.00 −0.772858
\(494\) 0 0
\(495\) 1296.00 0.117679
\(496\) 2624.00i 0.237542i
\(497\) 2640.00 0.238270
\(498\) 3312.00 0.298021
\(499\) − 8944.00i − 0.802382i −0.915995 0.401191i \(-0.868596\pi\)
0.915995 0.401191i \(-0.131404\pi\)
\(500\) − 5136.00i − 0.459378i
\(501\) 3348.00i 0.298558i
\(502\) − 4296.00i − 0.381952i
\(503\) −6528.00 −0.578666 −0.289333 0.957228i \(-0.593434\pi\)
−0.289333 + 0.957228i \(0.593434\pi\)
\(504\) −1440.00 −0.127267
\(505\) − 4788.00i − 0.421907i
\(506\) 3456.00 0.303632
\(507\) 0 0
\(508\) 8576.00 0.749013
\(509\) − 12114.0i − 1.05490i −0.849586 0.527450i \(-0.823148\pi\)
0.849586 0.527450i \(-0.176852\pi\)
\(510\) 1080.00 0.0937710
\(511\) −4360.00 −0.377446
\(512\) 512.000i 0.0441942i
\(513\) 432.000i 0.0371799i
\(514\) 1116.00i 0.0957678i
\(515\) 3120.00i 0.266958i
\(516\) −1968.00 −0.167900
\(517\) 4896.00 0.416491
\(518\) − 4400.00i − 0.373214i
\(519\) 13122.0 1.10981
\(520\) 0 0
\(521\) −14310.0 −1.20333 −0.601663 0.798750i \(-0.705495\pi\)
−0.601663 + 0.798750i \(0.705495\pi\)
\(522\) − 5076.00i − 0.425614i
\(523\) −18340.0 −1.53337 −0.766685 0.642024i \(-0.778095\pi\)
−0.766685 + 0.642024i \(0.778095\pi\)
\(524\) 10992.0 0.916389
\(525\) 5340.00i 0.443918i
\(526\) 4224.00i 0.350143i
\(527\) 4920.00i 0.406677i
\(528\) 1152.00i 0.0949514i
\(529\) −6983.00 −0.573929
\(530\) 8856.00 0.725811
\(531\) − 1080.00i − 0.0882637i
\(532\) 1280.00 0.104314
\(533\) 0 0
\(534\) −1260.00 −0.102108
\(535\) 72.0000i 0.00581838i
\(536\) 6784.00 0.546687
\(537\) 36.0000 0.00289295
\(538\) 10092.0i 0.808731i
\(539\) 1368.00i 0.109321i
\(540\) 648.000i 0.0516398i
\(541\) − 9254.00i − 0.735417i −0.929941 0.367708i \(-0.880142\pi\)
0.929941 0.367708i \(-0.119858\pi\)
\(542\) −7592.00 −0.601668
\(543\) 14154.0 1.11861
\(544\) 960.000i 0.0756611i
\(545\) 11004.0 0.864880
\(546\) 0 0
\(547\) 17444.0 1.36353 0.681766 0.731571i \(-0.261212\pi\)
0.681766 + 0.731571i \(0.261212\pi\)
\(548\) 11016.0i 0.858723i
\(549\) 5526.00 0.429588
\(550\) 4272.00 0.331198
\(551\) 4512.00i 0.348852i
\(552\) 1728.00i 0.133240i
\(553\) 21920.0i 1.68559i
\(554\) − 11164.0i − 0.856160i
\(555\) −1980.00 −0.151435
\(556\) −9008.00 −0.687094
\(557\) 3714.00i 0.282526i 0.989972 + 0.141263i \(0.0451164\pi\)
−0.989972 + 0.141263i \(0.954884\pi\)
\(558\) −2952.00 −0.223957
\(559\) 0 0
\(560\) 1920.00 0.144884
\(561\) 2160.00i 0.162558i
\(562\) −3900.00 −0.292725
\(563\) 13812.0 1.03394 0.516968 0.856004i \(-0.327060\pi\)
0.516968 + 0.856004i \(0.327060\pi\)
\(564\) 2448.00i 0.182765i
\(565\) − 2196.00i − 0.163516i
\(566\) 9464.00i 0.702830i
\(567\) − 1620.00i − 0.119989i
\(568\) 1056.00 0.0780084
\(569\) 15942.0 1.17456 0.587279 0.809385i \(-0.300199\pi\)
0.587279 + 0.809385i \(0.300199\pi\)
\(570\) − 576.000i − 0.0423263i
\(571\) −1604.00 −0.117557 −0.0587787 0.998271i \(-0.518721\pi\)
−0.0587787 + 0.998271i \(0.518721\pi\)
\(572\) 0 0
\(573\) 4104.00 0.299210
\(574\) − 5040.00i − 0.366490i
\(575\) 6408.00 0.464751
\(576\) −576.000 −0.0416667
\(577\) − 10654.0i − 0.768686i −0.923190 0.384343i \(-0.874428\pi\)
0.923190 0.384343i \(-0.125572\pi\)
\(578\) − 8026.00i − 0.577574i
\(579\) − 9930.00i − 0.712740i
\(580\) 6768.00i 0.484527i
\(581\) 11040.0 0.788324
\(582\) −10356.0 −0.737577
\(583\) 17712.0i 1.25824i
\(584\) −1744.00 −0.123574
\(585\) 0 0
\(586\) 9996.00 0.704660
\(587\) − 9984.00i − 0.702017i −0.936372 0.351008i \(-0.885839\pi\)
0.936372 0.351008i \(-0.114161\pi\)
\(588\) −684.000 −0.0479723
\(589\) 2624.00 0.183565
\(590\) 1440.00i 0.100481i
\(591\) − 9378.00i − 0.652723i
\(592\) − 1760.00i − 0.122188i
\(593\) − 12618.0i − 0.873793i −0.899512 0.436896i \(-0.856078\pi\)
0.899512 0.436896i \(-0.143922\pi\)
\(594\) −1296.00 −0.0895211
\(595\) 3600.00 0.248043
\(596\) 7080.00i 0.486591i
\(597\) 13992.0 0.959220
\(598\) 0 0
\(599\) 11184.0 0.762881 0.381441 0.924393i \(-0.375428\pi\)
0.381441 + 0.924393i \(0.375428\pi\)
\(600\) 2136.00i 0.145336i
\(601\) 2810.00 0.190719 0.0953596 0.995443i \(-0.469600\pi\)
0.0953596 + 0.995443i \(0.469600\pi\)
\(602\) −6560.00 −0.444129
\(603\) 7632.00i 0.515421i
\(604\) − 3952.00i − 0.266233i
\(605\) 4530.00i 0.304414i
\(606\) 4788.00i 0.320956i
\(607\) 1064.00 0.0711473 0.0355737 0.999367i \(-0.488674\pi\)
0.0355737 + 0.999367i \(0.488674\pi\)
\(608\) 512.000 0.0341519
\(609\) − 16920.0i − 1.12583i
\(610\) −7368.00 −0.489052
\(611\) 0 0
\(612\) −1080.00 −0.0713340
\(613\) − 20914.0i − 1.37799i −0.724766 0.688996i \(-0.758052\pi\)
0.724766 0.688996i \(-0.241948\pi\)
\(614\) 13648.0 0.897050
\(615\) −2268.00 −0.148707
\(616\) 3840.00i 0.251166i
\(617\) 9714.00i 0.633826i 0.948455 + 0.316913i \(0.102646\pi\)
−0.948455 + 0.316913i \(0.897354\pi\)
\(618\) − 3120.00i − 0.203082i
\(619\) 14848.0i 0.964122i 0.876138 + 0.482061i \(0.160112\pi\)
−0.876138 + 0.482061i \(0.839888\pi\)
\(620\) 3936.00 0.254957
\(621\) −1944.00 −0.125620
\(622\) 17520.0i 1.12940i
\(623\) −4200.00 −0.270095
\(624\) 0 0
\(625\) 3421.00 0.218944
\(626\) 7924.00i 0.505921i
\(627\) 1152.00 0.0733755
\(628\) −1304.00 −0.0828587
\(629\) − 3300.00i − 0.209189i
\(630\) 2160.00i 0.136598i
\(631\) − 19172.0i − 1.20955i −0.796397 0.604774i \(-0.793263\pi\)
0.796397 0.604774i \(-0.206737\pi\)
\(632\) 8768.00i 0.551855i
\(633\) 1668.00 0.104735
\(634\) −14172.0 −0.887763
\(635\) − 12864.0i − 0.803925i
\(636\) −8856.00 −0.552143
\(637\) 0 0
\(638\) −13536.0 −0.839961
\(639\) 1188.00i 0.0735470i
\(640\) 768.000 0.0474342
\(641\) 11502.0 0.708739 0.354369 0.935105i \(-0.384696\pi\)
0.354369 + 0.935105i \(0.384696\pi\)
\(642\) − 72.0000i − 0.00442619i
\(643\) − 15568.0i − 0.954809i −0.878684 0.477404i \(-0.841578\pi\)
0.878684 0.477404i \(-0.158422\pi\)
\(644\) 5760.00i 0.352447i
\(645\) 2952.00i 0.180209i
\(646\) 960.000 0.0584686
\(647\) −1128.00 −0.0685414 −0.0342707 0.999413i \(-0.510911\pi\)
−0.0342707 + 0.999413i \(0.510911\pi\)
\(648\) − 648.000i − 0.0392837i
\(649\) −2880.00 −0.174191
\(650\) 0 0
\(651\) −9840.00 −0.592412
\(652\) 5984.00i 0.359435i
\(653\) 8118.00 0.486496 0.243248 0.969964i \(-0.421787\pi\)
0.243248 + 0.969964i \(0.421787\pi\)
\(654\) −11004.0 −0.657936
\(655\) − 16488.0i − 0.983572i
\(656\) − 2016.00i − 0.119987i
\(657\) − 1962.00i − 0.116507i
\(658\) 8160.00i 0.483450i
\(659\) 13572.0 0.802261 0.401131 0.916021i \(-0.368617\pi\)
0.401131 + 0.916021i \(0.368617\pi\)
\(660\) 1728.00 0.101913
\(661\) 13138.0i 0.773085i 0.922272 + 0.386542i \(0.126331\pi\)
−0.922272 + 0.386542i \(0.873669\pi\)
\(662\) 18032.0 1.05866
\(663\) 0 0
\(664\) 4416.00 0.258093
\(665\) − 1920.00i − 0.111962i
\(666\) 1980.00 0.115200
\(667\) −20304.0 −1.17867
\(668\) 4464.00i 0.258559i
\(669\) 804.000i 0.0464640i
\(670\) − 10176.0i − 0.586766i
\(671\) − 14736.0i − 0.847805i
\(672\) −1920.00 −0.110217
\(673\) 718.000 0.0411246 0.0205623 0.999789i \(-0.493454\pi\)
0.0205623 + 0.999789i \(0.493454\pi\)
\(674\) − 4612.00i − 0.263572i
\(675\) −2403.00 −0.137024
\(676\) 0 0
\(677\) −2994.00 −0.169969 −0.0849843 0.996382i \(-0.527084\pi\)
−0.0849843 + 0.996382i \(0.527084\pi\)
\(678\) 2196.00i 0.124391i
\(679\) −34520.0 −1.95104
\(680\) 1440.00 0.0812081
\(681\) − 5400.00i − 0.303860i
\(682\) 7872.00i 0.441986i
\(683\) − 27384.0i − 1.53414i −0.641562 0.767071i \(-0.721713\pi\)
0.641562 0.767071i \(-0.278287\pi\)
\(684\) 576.000i 0.0321987i
\(685\) 16524.0 0.921678
\(686\) 11440.0 0.636707
\(687\) 8970.00i 0.498147i
\(688\) −2624.00 −0.145406
\(689\) 0 0
\(690\) 2592.00 0.143008
\(691\) 27632.0i 1.52123i 0.649202 + 0.760616i \(0.275103\pi\)
−0.649202 + 0.760616i \(0.724897\pi\)
\(692\) 17496.0 0.961124
\(693\) −4320.00 −0.236801
\(694\) − 22152.0i − 1.21164i
\(695\) 13512.0i 0.737467i
\(696\) − 6768.00i − 0.368592i
\(697\) − 3780.00i − 0.205420i
\(698\) 4684.00 0.254000
\(699\) 8478.00 0.458752
\(700\) 7120.00i 0.384444i
\(701\) −19062.0 −1.02705 −0.513525 0.858075i \(-0.671661\pi\)
−0.513525 + 0.858075i \(0.671661\pi\)
\(702\) 0 0
\(703\) −1760.00 −0.0944234
\(704\) 1536.00i 0.0822304i
\(705\) 3672.00 0.196164
\(706\) −9300.00 −0.495765
\(707\) 15960.0i 0.848992i
\(708\) − 1440.00i − 0.0764386i
\(709\) − 3854.00i − 0.204147i −0.994777 0.102073i \(-0.967452\pi\)
0.994777 0.102073i \(-0.0325476\pi\)
\(710\) − 1584.00i − 0.0837274i
\(711\) −9864.00 −0.520294
\(712\) −1680.00 −0.0884279
\(713\) 11808.0i 0.620215i
\(714\) −3600.00 −0.188693
\(715\) 0 0
\(716\) 48.0000 0.00250537
\(717\) 5436.00i 0.283140i
\(718\) −22536.0 −1.17136
\(719\) −20976.0 −1.08800 −0.544001 0.839085i \(-0.683091\pi\)
−0.544001 + 0.839085i \(0.683091\pi\)
\(720\) 864.000i 0.0447214i
\(721\) − 10400.0i − 0.537193i
\(722\) 13206.0i 0.680715i
\(723\) − 4746.00i − 0.244130i
\(724\) 18872.0 0.968746
\(725\) −25098.0 −1.28568
\(726\) − 4530.00i − 0.231576i
\(727\) 29464.0 1.50311 0.751554 0.659672i \(-0.229305\pi\)
0.751554 + 0.659672i \(0.229305\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 2616.00i 0.132634i
\(731\) −4920.00 −0.248937
\(732\) 7368.00 0.372034
\(733\) − 2698.00i − 0.135952i −0.997687 0.0679761i \(-0.978346\pi\)
0.997687 0.0679761i \(-0.0216542\pi\)
\(734\) − 14576.0i − 0.732984i
\(735\) 1026.00i 0.0514892i
\(736\) 2304.00i 0.115389i
\(737\) 20352.0 1.01720
\(738\) 2268.00 0.113125
\(739\) − 632.000i − 0.0314594i −0.999876 0.0157297i \(-0.994993\pi\)
0.999876 0.0157297i \(-0.00500713\pi\)
\(740\) −2640.00 −0.131146
\(741\) 0 0
\(742\) −29520.0 −1.46053
\(743\) − 20844.0i − 1.02920i −0.857432 0.514598i \(-0.827941\pi\)
0.857432 0.514598i \(-0.172059\pi\)
\(744\) −3936.00 −0.193953
\(745\) 10620.0 0.522264
\(746\) − 19940.0i − 0.978626i
\(747\) 4968.00i 0.243333i
\(748\) 2880.00i 0.140780i
\(749\) − 240.000i − 0.0117082i
\(750\) 7704.00 0.375080
\(751\) −272.000 −0.0132163 −0.00660814 0.999978i \(-0.502103\pi\)
−0.00660814 + 0.999978i \(0.502103\pi\)
\(752\) 3264.00i 0.158279i
\(753\) 6444.00 0.311862
\(754\) 0 0
\(755\) −5928.00 −0.285751
\(756\) − 2160.00i − 0.103913i
\(757\) 37550.0 1.80288 0.901439 0.432907i \(-0.142512\pi\)
0.901439 + 0.432907i \(0.142512\pi\)
\(758\) −26896.0 −1.28880
\(759\) 5184.00i 0.247915i
\(760\) − 768.000i − 0.0366556i
\(761\) − 33330.0i − 1.58766i −0.608138 0.793832i \(-0.708083\pi\)
0.608138 0.793832i \(-0.291917\pi\)
\(762\) 12864.0i 0.611566i
\(763\) −36680.0 −1.74037
\(764\) 5472.00 0.259123
\(765\) 1620.00i 0.0765637i
\(766\) −23640.0 −1.11508
\(767\) 0 0
\(768\) −768.000 −0.0360844
\(769\) − 15406.0i − 0.722438i −0.932481 0.361219i \(-0.882361\pi\)
0.932481 0.361219i \(-0.117639\pi\)
\(770\) 5760.00 0.269579
\(771\) −1674.00 −0.0781941
\(772\) − 13240.0i − 0.617251i
\(773\) − 29514.0i − 1.37328i −0.726998 0.686640i \(-0.759085\pi\)
0.726998 0.686640i \(-0.240915\pi\)
\(774\) − 2952.00i − 0.137090i
\(775\) 14596.0i 0.676521i
\(776\) −13808.0 −0.638761
\(777\) 6600.00 0.304728
\(778\) − 348.000i − 0.0160365i
\(779\) −2016.00 −0.0927223
\(780\) 0 0
\(781\) 3168.00 0.145147
\(782\) 4320.00i 0.197548i
\(783\) 7614.00 0.347512
\(784\) −912.000 −0.0415452
\(785\) 1956.00i 0.0889333i
\(786\) 16488.0i 0.748228i
\(787\) − 33176.0i − 1.50266i −0.659924 0.751332i \(-0.729412\pi\)
0.659924 0.751332i \(-0.270588\pi\)
\(788\) − 12504.0i − 0.565275i
\(789\) −6336.00 −0.285890
\(790\) 13152.0 0.592313
\(791\) 7320.00i 0.329038i
\(792\) −1728.00 −0.0775275
\(793\) 0 0
\(794\) −5972.00 −0.266925
\(795\) 13284.0i 0.592623i
\(796\) 18656.0 0.830709
\(797\) 16746.0 0.744258 0.372129 0.928181i \(-0.378628\pi\)
0.372129 + 0.928181i \(0.378628\pi\)
\(798\) 1920.00i 0.0851720i
\(799\) 6120.00i 0.270976i
\(800\) 2848.00i 0.125865i
\(801\) − 1890.00i − 0.0833706i
\(802\) −21132.0 −0.930420
\(803\) −5232.00 −0.229929
\(804\) 10176.0i 0.446368i
\(805\) 8640.00 0.378286
\(806\) 0 0
\(807\) −15138.0 −0.660326
\(808\) 6384.00i 0.277956i
\(809\) −15846.0 −0.688647 −0.344324 0.938851i \(-0.611892\pi\)
−0.344324 + 0.938851i \(0.611892\pi\)
\(810\) −972.000 −0.0421637
\(811\) 22952.0i 0.993778i 0.867814 + 0.496889i \(0.165524\pi\)
−0.867814 + 0.496889i \(0.834476\pi\)
\(812\) − 22560.0i − 0.975001i
\(813\) − 11388.0i − 0.491260i
\(814\) − 5280.00i − 0.227351i
\(815\) 8976.00 0.385786
\(816\) −1440.00 −0.0617771
\(817\) 2624.00i 0.112365i
\(818\) 14540.0 0.621490
\(819\) 0 0
\(820\) −3024.00 −0.128784
\(821\) − 37146.0i − 1.57906i −0.613715 0.789528i \(-0.710326\pi\)
0.613715 0.789528i \(-0.289674\pi\)
\(822\) −16524.0 −0.701144
\(823\) 9592.00 0.406265 0.203133 0.979151i \(-0.434888\pi\)
0.203133 + 0.979151i \(0.434888\pi\)
\(824\) − 4160.00i − 0.175874i
\(825\) 6408.00i 0.270422i
\(826\) − 4800.00i − 0.202195i
\(827\) 39960.0i 1.68022i 0.542413 + 0.840112i \(0.317511\pi\)
−0.542413 + 0.840112i \(0.682489\pi\)
\(828\) −2592.00 −0.108790
\(829\) 3706.00 0.155265 0.0776325 0.996982i \(-0.475264\pi\)
0.0776325 + 0.996982i \(0.475264\pi\)
\(830\) − 6624.00i − 0.277015i
\(831\) 16746.0 0.699052
\(832\) 0 0
\(833\) −1710.00 −0.0711260
\(834\) − 13512.0i − 0.561010i
\(835\) 6696.00 0.277515
\(836\) 1536.00 0.0635451
\(837\) − 4428.00i − 0.182860i
\(838\) − 14616.0i − 0.602508i
\(839\) − 9756.00i − 0.401448i −0.979648 0.200724i \(-0.935671\pi\)
0.979648 0.200724i \(-0.0643294\pi\)
\(840\) 2880.00i 0.118297i
\(841\) 55135.0 2.26065
\(842\) 11876.0 0.486074
\(843\) − 5850.00i − 0.239009i
\(844\) 2224.00 0.0907029
\(845\) 0 0
\(846\) −3672.00 −0.149227
\(847\) − 15100.0i − 0.612565i
\(848\) −11808.0 −0.478170
\(849\) −14196.0 −0.573858
\(850\) 5340.00i 0.215483i
\(851\) − 7920.00i − 0.319029i
\(852\) 1584.00i 0.0636936i
\(853\) − 11342.0i − 0.455267i −0.973747 0.227633i \(-0.926901\pi\)
0.973747 0.227633i \(-0.0730988\pi\)
\(854\) 24560.0 0.984105
\(855\) 864.000 0.0345593
\(856\) − 96.0000i − 0.00383319i
\(857\) 16134.0 0.643089 0.321544 0.946895i \(-0.395798\pi\)
0.321544 + 0.946895i \(0.395798\pi\)
\(858\) 0 0
\(859\) −20932.0 −0.831421 −0.415710 0.909497i \(-0.636467\pi\)
−0.415710 + 0.909497i \(0.636467\pi\)
\(860\) 3936.00i 0.156066i
\(861\) 7560.00 0.299238
\(862\) −23064.0 −0.911326
\(863\) 10044.0i 0.396178i 0.980184 + 0.198089i \(0.0634735\pi\)
−0.980184 + 0.198089i \(0.936526\pi\)
\(864\) − 864.000i − 0.0340207i
\(865\) − 26244.0i − 1.03159i
\(866\) 1436.00i 0.0563479i
\(867\) 12039.0 0.471587
\(868\) −13120.0 −0.513044
\(869\) 26304.0i 1.02681i
\(870\) −10152.0 −0.395615
\(871\) 0 0
\(872\) −14672.0 −0.569790
\(873\) − 15534.0i − 0.602229i
\(874\) 2304.00 0.0891693
\(875\) 25680.0 0.992163
\(876\) − 2616.00i − 0.100898i
\(877\) − 26314.0i − 1.01318i −0.862186 0.506591i \(-0.830905\pi\)
0.862186 0.506591i \(-0.169095\pi\)
\(878\) − 17968.0i − 0.690650i
\(879\) 14994.0i 0.575353i
\(880\) 2304.00 0.0882589
\(881\) −37506.0 −1.43429 −0.717145 0.696924i \(-0.754551\pi\)
−0.717145 + 0.696924i \(0.754551\pi\)
\(882\) − 1026.00i − 0.0391692i
\(883\) 6388.00 0.243458 0.121729 0.992563i \(-0.461156\pi\)
0.121729 + 0.992563i \(0.461156\pi\)
\(884\) 0 0
\(885\) −2160.00 −0.0820425
\(886\) 5208.00i 0.197479i
\(887\) −5472.00 −0.207138 −0.103569 0.994622i \(-0.533026\pi\)
−0.103569 + 0.994622i \(0.533026\pi\)
\(888\) 2640.00 0.0997664
\(889\) 42880.0i 1.61772i
\(890\) 2520.00i 0.0949108i
\(891\) − 1944.00i − 0.0730937i
\(892\) 1072.00i 0.0402390i
\(893\) 3264.00 0.122313
\(894\) −10620.0 −0.397300
\(895\) − 72.0000i − 0.00268904i
\(896\) −2560.00 −0.0954504
\(897\) 0 0
\(898\) −26412.0 −0.981492
\(899\) − 46248.0i − 1.71575i
\(900\) −3204.00 −0.118667
\(901\) −22140.0 −0.818635
\(902\) − 6048.00i − 0.223255i
\(903\) − 9840.00i − 0.362630i
\(904\) 2928.00i 0.107725i
\(905\) − 28308.0i − 1.03977i
\(906\) 5928.00 0.217378
\(907\) 7180.00 0.262853 0.131427 0.991326i \(-0.458044\pi\)
0.131427 + 0.991326i \(0.458044\pi\)
\(908\) − 7200.00i − 0.263150i
\(909\) −7182.00 −0.262059
\(910\) 0 0
\(911\) 27624.0 1.00464 0.502318 0.864683i \(-0.332481\pi\)
0.502318 + 0.864683i \(0.332481\pi\)
\(912\) 768.000i 0.0278849i
\(913\) 13248.0 0.480224
\(914\) −16852.0 −0.609863
\(915\) − 11052.0i − 0.399309i
\(916\) 11960.0i 0.431408i
\(917\) 54960.0i 1.97921i
\(918\) − 1620.00i − 0.0582440i
\(919\) −30256.0 −1.08602 −0.543011 0.839726i \(-0.682716\pi\)
−0.543011 + 0.839726i \(0.682716\pi\)
\(920\) 3456.00 0.123849
\(921\) 20472.0i 0.732438i
\(922\) −33372.0 −1.19203
\(923\) 0 0
\(924\) −5760.00 −0.205076
\(925\) − 9790.00i − 0.347993i
\(926\) 31864.0 1.13079
\(927\) 4680.00 0.165816
\(928\) − 9024.00i − 0.319210i
\(929\) − 1926.00i − 0.0680194i −0.999422 0.0340097i \(-0.989172\pi\)
0.999422 0.0340097i \(-0.0108277\pi\)
\(930\) 5904.00i 0.208172i
\(931\) 912.000i 0.0321048i
\(932\) 11304.0 0.397291
\(933\) −26280.0 −0.922153
\(934\) − 37080.0i − 1.29903i
\(935\) 4320.00 0.151101
\(936\) 0 0
\(937\) 3962.00 0.138135 0.0690677 0.997612i \(-0.477998\pi\)
0.0690677 + 0.997612i \(0.477998\pi\)
\(938\) 33920.0i 1.18073i
\(939\) −11886.0 −0.413083
\(940\) 4896.00 0.169883
\(941\) − 1074.00i − 0.0372066i −0.999827 0.0186033i \(-0.994078\pi\)
0.999827 0.0186033i \(-0.00592195\pi\)
\(942\) − 1956.00i − 0.0676538i
\(943\) − 9072.00i − 0.313282i
\(944\) − 1920.00i − 0.0661978i
\(945\) −3240.00 −0.111531
\(946\) −7872.00 −0.270551
\(947\) − 4848.00i − 0.166356i −0.996535 0.0831778i \(-0.973493\pi\)
0.996535 0.0831778i \(-0.0265070\pi\)
\(948\) −13152.0 −0.450588
\(949\) 0 0
\(950\) 2848.00 0.0972645
\(951\) − 21258.0i − 0.724856i
\(952\) −4800.00 −0.163413
\(953\) −762.000 −0.0259009 −0.0129505 0.999916i \(-0.504122\pi\)
−0.0129505 + 0.999916i \(0.504122\pi\)
\(954\) − 13284.0i − 0.450823i
\(955\) − 8208.00i − 0.278120i
\(956\) 7248.00i 0.245206i
\(957\) − 20304.0i − 0.685826i
\(958\) 12360.0 0.416841
\(959\) −55080.0 −1.85467
\(960\) 1152.00i 0.0387298i
\(961\) 2895.00 0.0971770
\(962\) 0 0
\(963\) 108.000 0.00361397
\(964\) − 6328.00i − 0.211422i
\(965\) −19860.0 −0.662504
\(966\) −8640.00 −0.287772
\(967\) 35804.0i 1.19067i 0.803477 + 0.595336i \(0.202981\pi\)
−0.803477 + 0.595336i \(0.797019\pi\)
\(968\) − 6040.00i − 0.200551i
\(969\) 1440.00i 0.0477394i
\(970\) 20712.0i 0.685590i
\(971\) −4260.00 −0.140793 −0.0703964 0.997519i \(-0.522426\pi\)
−0.0703964 + 0.997519i \(0.522426\pi\)
\(972\) 972.000 0.0320750
\(973\) − 45040.0i − 1.48398i
\(974\) −23512.0 −0.773484
\(975\) 0 0
\(976\) 9824.00 0.322191
\(977\) − 28710.0i − 0.940137i −0.882630 0.470069i \(-0.844229\pi\)
0.882630 0.470069i \(-0.155771\pi\)
\(978\) −8976.00 −0.293477
\(979\) −5040.00 −0.164534
\(980\) 1368.00i 0.0445910i
\(981\) − 16506.0i − 0.537203i
\(982\) − 3816.00i − 0.124006i
\(983\) 49524.0i 1.60689i 0.595381 + 0.803444i \(0.297001\pi\)
−0.595381 + 0.803444i \(0.702999\pi\)
\(984\) 3024.00 0.0979691
\(985\) −18756.0 −0.606717
\(986\) − 16920.0i − 0.546493i
\(987\) −12240.0 −0.394735
\(988\) 0 0
\(989\) −11808.0 −0.379649
\(990\) 2592.00i 0.0832113i
\(991\) 44408.0 1.42348 0.711739 0.702444i \(-0.247908\pi\)
0.711739 + 0.702444i \(0.247908\pi\)
\(992\) −5248.00 −0.167968
\(993\) 27048.0i 0.864393i
\(994\) 5280.00i 0.168482i
\(995\) − 27984.0i − 0.891610i
\(996\) 6624.00i 0.210732i
\(997\) 18398.0 0.584424 0.292212 0.956354i \(-0.405609\pi\)
0.292212 + 0.956354i \(0.405609\pi\)
\(998\) 17888.0 0.567369
\(999\) 2970.00i 0.0940607i
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 1014.4.b.c.337.2 2
13.5 odd 4 78.4.a.e.1.1 1
13.8 odd 4 1014.4.a.b.1.1 1
13.12 even 2 inner 1014.4.b.c.337.1 2
39.5 even 4 234.4.a.b.1.1 1
52.31 even 4 624.4.a.i.1.1 1
65.44 odd 4 1950.4.a.c.1.1 1
104.5 odd 4 2496.4.a.k.1.1 1
104.83 even 4 2496.4.a.b.1.1 1
156.83 odd 4 1872.4.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.e.1.1 1 13.5 odd 4
234.4.a.b.1.1 1 39.5 even 4
624.4.a.i.1.1 1 52.31 even 4
1014.4.a.b.1.1 1 13.8 odd 4
1014.4.b.c.337.1 2 13.12 even 2 inner
1014.4.b.c.337.2 2 1.1 even 1 trivial
1872.4.a.e.1.1 1 156.83 odd 4
1950.4.a.c.1.1 1 65.44 odd 4
2496.4.a.b.1.1 1 104.83 even 4
2496.4.a.k.1.1 1 104.5 odd 4