Properties

Label 21.3.d.a.13.1
Level $21$
Weight $3$
Character 21.13
Analytic conductor $0.572$
Analytic rank $0$
Dimension $2$
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] = [21,3,Mod(13,21)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(21, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.13");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 21.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.572208555157\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 13.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 21.13
Dual form 21.3.d.a.13.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.73205i q^{3} -3.00000 q^{4} +6.92820i q^{5} -1.73205i q^{6} +(1.00000 - 6.92820i) q^{7} -7.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.73205i q^{3} -3.00000 q^{4} +6.92820i q^{5} -1.73205i q^{6} +(1.00000 - 6.92820i) q^{7} -7.00000 q^{8} -3.00000 q^{9} +6.92820i q^{10} +10.0000 q^{11} +5.19615i q^{12} -6.92820i q^{13} +(1.00000 - 6.92820i) q^{14} +12.0000 q^{15} +5.00000 q^{16} -3.00000 q^{18} +20.7846i q^{19} -20.7846i q^{20} +(-12.0000 - 1.73205i) q^{21} +10.0000 q^{22} -14.0000 q^{23} +12.1244i q^{24} -23.0000 q^{25} -6.92820i q^{26} +5.19615i q^{27} +(-3.00000 + 20.7846i) q^{28} -38.0000 q^{29} +12.0000 q^{30} -27.7128i q^{31} +33.0000 q^{32} -17.3205i q^{33} +(48.0000 + 6.92820i) q^{35} +9.00000 q^{36} +26.0000 q^{37} +20.7846i q^{38} -12.0000 q^{39} -48.4974i q^{40} +69.2820i q^{41} +(-12.0000 - 1.73205i) q^{42} +26.0000 q^{43} -30.0000 q^{44} -20.7846i q^{45} -14.0000 q^{46} -27.7128i q^{47} -8.66025i q^{48} +(-47.0000 - 13.8564i) q^{49} -23.0000 q^{50} +20.7846i q^{52} +10.0000 q^{53} +5.19615i q^{54} +69.2820i q^{55} +(-7.00000 + 48.4974i) q^{56} +36.0000 q^{57} -38.0000 q^{58} -76.2102i q^{59} -36.0000 q^{60} -34.6410i q^{61} -27.7128i q^{62} +(-3.00000 + 20.7846i) q^{63} +13.0000 q^{64} +48.0000 q^{65} -17.3205i q^{66} +74.0000 q^{67} +24.2487i q^{69} +(48.0000 + 6.92820i) q^{70} -62.0000 q^{71} +21.0000 q^{72} +41.5692i q^{73} +26.0000 q^{74} +39.8372i q^{75} -62.3538i q^{76} +(10.0000 - 69.2820i) q^{77} -12.0000 q^{78} -46.0000 q^{79} +34.6410i q^{80} +9.00000 q^{81} +69.2820i q^{82} -90.0666i q^{83} +(36.0000 + 5.19615i) q^{84} +26.0000 q^{86} +65.8179i q^{87} -70.0000 q^{88} +41.5692i q^{89} -20.7846i q^{90} +(-48.0000 - 6.92820i) q^{91} +42.0000 q^{92} -48.0000 q^{93} -27.7128i q^{94} -144.000 q^{95} -57.1577i q^{96} -55.4256i q^{97} +(-47.0000 - 13.8564i) q^{98} -30.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 6 q^{4} + 2 q^{7} - 14 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 6 q^{4} + 2 q^{7} - 14 q^{8} - 6 q^{9} + 20 q^{11} + 2 q^{14} + 24 q^{15} + 10 q^{16} - 6 q^{18} - 24 q^{21} + 20 q^{22} - 28 q^{23} - 46 q^{25} - 6 q^{28} - 76 q^{29} + 24 q^{30} + 66 q^{32} + 96 q^{35} + 18 q^{36} + 52 q^{37} - 24 q^{39} - 24 q^{42} + 52 q^{43} - 60 q^{44} - 28 q^{46} - 94 q^{49} - 46 q^{50} + 20 q^{53} - 14 q^{56} + 72 q^{57} - 76 q^{58} - 72 q^{60} - 6 q^{63} + 26 q^{64} + 96 q^{65} + 148 q^{67} + 96 q^{70} - 124 q^{71} + 42 q^{72} + 52 q^{74} + 20 q^{77} - 24 q^{78} - 92 q^{79} + 18 q^{81} + 72 q^{84} + 52 q^{86} - 140 q^{88} - 96 q^{91} + 84 q^{92} - 96 q^{93} - 288 q^{95} - 94 q^{98} - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\) \(8\) \(10\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.500000 0.250000 0.968246i \(-0.419569\pi\)
0.250000 + 0.968246i \(0.419569\pi\)
\(3\) 1.73205i 0.577350i
\(4\) −3.00000 −0.750000
\(5\) 6.92820i 1.38564i 0.721110 + 0.692820i \(0.243632\pi\)
−0.721110 + 0.692820i \(0.756368\pi\)
\(6\) 1.73205i 0.288675i
\(7\) 1.00000 6.92820i 0.142857 0.989743i
\(8\) −7.00000 −0.875000
\(9\) −3.00000 −0.333333
\(10\) 6.92820i 0.692820i
\(11\) 10.0000 0.909091 0.454545 0.890724i \(-0.349802\pi\)
0.454545 + 0.890724i \(0.349802\pi\)
\(12\) 5.19615i 0.433013i
\(13\) 6.92820i 0.532939i −0.963843 0.266469i \(-0.914143\pi\)
0.963843 0.266469i \(-0.0858571\pi\)
\(14\) 1.00000 6.92820i 0.0714286 0.494872i
\(15\) 12.0000 0.800000
\(16\) 5.00000 0.312500
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −3.00000 −0.166667
\(19\) 20.7846i 1.09393i 0.837157 + 0.546963i \(0.184216\pi\)
−0.837157 + 0.546963i \(0.815784\pi\)
\(20\) 20.7846i 1.03923i
\(21\) −12.0000 1.73205i −0.571429 0.0824786i
\(22\) 10.0000 0.454545
\(23\) −14.0000 −0.608696 −0.304348 0.952561i \(-0.598439\pi\)
−0.304348 + 0.952561i \(0.598439\pi\)
\(24\) 12.1244i 0.505181i
\(25\) −23.0000 −0.920000
\(26\) 6.92820i 0.266469i
\(27\) 5.19615i 0.192450i
\(28\) −3.00000 + 20.7846i −0.107143 + 0.742307i
\(29\) −38.0000 −1.31034 −0.655172 0.755479i \(-0.727404\pi\)
−0.655172 + 0.755479i \(0.727404\pi\)
\(30\) 12.0000 0.400000
\(31\) 27.7128i 0.893962i −0.894544 0.446981i \(-0.852499\pi\)
0.894544 0.446981i \(-0.147501\pi\)
\(32\) 33.0000 1.03125
\(33\) 17.3205i 0.524864i
\(34\) 0 0
\(35\) 48.0000 + 6.92820i 1.37143 + 0.197949i
\(36\) 9.00000 0.250000
\(37\) 26.0000 0.702703 0.351351 0.936244i \(-0.385722\pi\)
0.351351 + 0.936244i \(0.385722\pi\)
\(38\) 20.7846i 0.546963i
\(39\) −12.0000 −0.307692
\(40\) 48.4974i 1.21244i
\(41\) 69.2820i 1.68981i 0.534920 + 0.844903i \(0.320342\pi\)
−0.534920 + 0.844903i \(0.679658\pi\)
\(42\) −12.0000 1.73205i −0.285714 0.0412393i
\(43\) 26.0000 0.604651 0.302326 0.953205i \(-0.402237\pi\)
0.302326 + 0.953205i \(0.402237\pi\)
\(44\) −30.0000 −0.681818
\(45\) 20.7846i 0.461880i
\(46\) −14.0000 −0.304348
\(47\) 27.7128i 0.589634i −0.955554 0.294817i \(-0.904741\pi\)
0.955554 0.294817i \(-0.0952587\pi\)
\(48\) 8.66025i 0.180422i
\(49\) −47.0000 13.8564i −0.959184 0.282784i
\(50\) −23.0000 −0.460000
\(51\) 0 0
\(52\) 20.7846i 0.399704i
\(53\) 10.0000 0.188679 0.0943396 0.995540i \(-0.469926\pi\)
0.0943396 + 0.995540i \(0.469926\pi\)
\(54\) 5.19615i 0.0962250i
\(55\) 69.2820i 1.25967i
\(56\) −7.00000 + 48.4974i −0.125000 + 0.866025i
\(57\) 36.0000 0.631579
\(58\) −38.0000 −0.655172
\(59\) 76.2102i 1.29170i −0.763465 0.645849i \(-0.776503\pi\)
0.763465 0.645849i \(-0.223497\pi\)
\(60\) −36.0000 −0.600000
\(61\) 34.6410i 0.567886i −0.958841 0.283943i \(-0.908357\pi\)
0.958841 0.283943i \(-0.0916426\pi\)
\(62\) 27.7128i 0.446981i
\(63\) −3.00000 + 20.7846i −0.0476190 + 0.329914i
\(64\) 13.0000 0.203125
\(65\) 48.0000 0.738462
\(66\) 17.3205i 0.262432i
\(67\) 74.0000 1.10448 0.552239 0.833686i \(-0.313774\pi\)
0.552239 + 0.833686i \(0.313774\pi\)
\(68\) 0 0
\(69\) 24.2487i 0.351431i
\(70\) 48.0000 + 6.92820i 0.685714 + 0.0989743i
\(71\) −62.0000 −0.873239 −0.436620 0.899646i \(-0.643824\pi\)
−0.436620 + 0.899646i \(0.643824\pi\)
\(72\) 21.0000 0.291667
\(73\) 41.5692i 0.569441i 0.958611 + 0.284721i \(0.0919009\pi\)
−0.958611 + 0.284721i \(0.908099\pi\)
\(74\) 26.0000 0.351351
\(75\) 39.8372i 0.531162i
\(76\) 62.3538i 0.820445i
\(77\) 10.0000 69.2820i 0.129870 0.899767i
\(78\) −12.0000 −0.153846
\(79\) −46.0000 −0.582278 −0.291139 0.956681i \(-0.594034\pi\)
−0.291139 + 0.956681i \(0.594034\pi\)
\(80\) 34.6410i 0.433013i
\(81\) 9.00000 0.111111
\(82\) 69.2820i 0.844903i
\(83\) 90.0666i 1.08514i −0.840011 0.542570i \(-0.817451\pi\)
0.840011 0.542570i \(-0.182549\pi\)
\(84\) 36.0000 + 5.19615i 0.428571 + 0.0618590i
\(85\) 0 0
\(86\) 26.0000 0.302326
\(87\) 65.8179i 0.756528i
\(88\) −70.0000 −0.795455
\(89\) 41.5692i 0.467070i 0.972348 + 0.233535i \(0.0750293\pi\)
−0.972348 + 0.233535i \(0.924971\pi\)
\(90\) 20.7846i 0.230940i
\(91\) −48.0000 6.92820i −0.527473 0.0761341i
\(92\) 42.0000 0.456522
\(93\) −48.0000 −0.516129
\(94\) 27.7128i 0.294817i
\(95\) −144.000 −1.51579
\(96\) 57.1577i 0.595392i
\(97\) 55.4256i 0.571398i −0.958319 0.285699i \(-0.907774\pi\)
0.958319 0.285699i \(-0.0922258\pi\)
\(98\) −47.0000 13.8564i −0.479592 0.141392i
\(99\) −30.0000 −0.303030
\(100\) 69.0000 0.690000
\(101\) 117.779i 1.16613i 0.812424 + 0.583067i \(0.198147\pi\)
−0.812424 + 0.583067i \(0.801853\pi\)
\(102\) 0 0
\(103\) 96.9948i 0.941698i 0.882214 + 0.470849i \(0.156052\pi\)
−0.882214 + 0.470849i \(0.843948\pi\)
\(104\) 48.4974i 0.466321i
\(105\) 12.0000 83.1384i 0.114286 0.791795i
\(106\) 10.0000 0.0943396
\(107\) 10.0000 0.0934579 0.0467290 0.998908i \(-0.485120\pi\)
0.0467290 + 0.998908i \(0.485120\pi\)
\(108\) 15.5885i 0.144338i
\(109\) 74.0000 0.678899 0.339450 0.940624i \(-0.389759\pi\)
0.339450 + 0.940624i \(0.389759\pi\)
\(110\) 69.2820i 0.629837i
\(111\) 45.0333i 0.405706i
\(112\) 5.00000 34.6410i 0.0446429 0.309295i
\(113\) 178.000 1.57522 0.787611 0.616173i \(-0.211318\pi\)
0.787611 + 0.616173i \(0.211318\pi\)
\(114\) 36.0000 0.315789
\(115\) 96.9948i 0.843433i
\(116\) 114.000 0.982759
\(117\) 20.7846i 0.177646i
\(118\) 76.2102i 0.645849i
\(119\) 0 0
\(120\) −84.0000 −0.700000
\(121\) −21.0000 −0.173554
\(122\) 34.6410i 0.283943i
\(123\) 120.000 0.975610
\(124\) 83.1384i 0.670471i
\(125\) 13.8564i 0.110851i
\(126\) −3.00000 + 20.7846i −0.0238095 + 0.164957i
\(127\) −190.000 −1.49606 −0.748031 0.663663i \(-0.769001\pi\)
−0.748031 + 0.663663i \(0.769001\pi\)
\(128\) −119.000 −0.929688
\(129\) 45.0333i 0.349096i
\(130\) 48.0000 0.369231
\(131\) 48.4974i 0.370209i 0.982719 + 0.185105i \(0.0592624\pi\)
−0.982719 + 0.185105i \(0.940738\pi\)
\(132\) 51.9615i 0.393648i
\(133\) 144.000 + 20.7846i 1.08271 + 0.156275i
\(134\) 74.0000 0.552239
\(135\) −36.0000 −0.266667
\(136\) 0 0
\(137\) −206.000 −1.50365 −0.751825 0.659363i \(-0.770826\pi\)
−0.751825 + 0.659363i \(0.770826\pi\)
\(138\) 24.2487i 0.175715i
\(139\) 117.779i 0.847334i 0.905818 + 0.423667i \(0.139257\pi\)
−0.905818 + 0.423667i \(0.860743\pi\)
\(140\) −144.000 20.7846i −1.02857 0.148461i
\(141\) −48.0000 −0.340426
\(142\) −62.0000 −0.436620
\(143\) 69.2820i 0.484490i
\(144\) −15.0000 −0.104167
\(145\) 263.272i 1.81567i
\(146\) 41.5692i 0.284721i
\(147\) −24.0000 + 81.4064i −0.163265 + 0.553785i
\(148\) −78.0000 −0.527027
\(149\) 106.000 0.711409 0.355705 0.934598i \(-0.384241\pi\)
0.355705 + 0.934598i \(0.384241\pi\)
\(150\) 39.8372i 0.265581i
\(151\) −46.0000 −0.304636 −0.152318 0.988332i \(-0.548674\pi\)
−0.152318 + 0.988332i \(0.548674\pi\)
\(152\) 145.492i 0.957186i
\(153\) 0 0
\(154\) 10.0000 69.2820i 0.0649351 0.449883i
\(155\) 192.000 1.23871
\(156\) 36.0000 0.230769
\(157\) 242.487i 1.54450i 0.635316 + 0.772252i \(0.280870\pi\)
−0.635316 + 0.772252i \(0.719130\pi\)
\(158\) −46.0000 −0.291139
\(159\) 17.3205i 0.108934i
\(160\) 228.631i 1.42894i
\(161\) −14.0000 + 96.9948i −0.0869565 + 0.602452i
\(162\) 9.00000 0.0555556
\(163\) 170.000 1.04294 0.521472 0.853268i \(-0.325383\pi\)
0.521472 + 0.853268i \(0.325383\pi\)
\(164\) 207.846i 1.26735i
\(165\) 120.000 0.727273
\(166\) 90.0666i 0.542570i
\(167\) 96.9948i 0.580807i −0.956904 0.290404i \(-0.906210\pi\)
0.956904 0.290404i \(-0.0937896\pi\)
\(168\) 84.0000 + 12.1244i 0.500000 + 0.0721688i
\(169\) 121.000 0.715976
\(170\) 0 0
\(171\) 62.3538i 0.364642i
\(172\) −78.0000 −0.453488
\(173\) 173.205i 1.00119i −0.865683 0.500593i \(-0.833115\pi\)
0.865683 0.500593i \(-0.166885\pi\)
\(174\) 65.8179i 0.378264i
\(175\) −23.0000 + 159.349i −0.131429 + 0.910564i
\(176\) 50.0000 0.284091
\(177\) −132.000 −0.745763
\(178\) 41.5692i 0.233535i
\(179\) −38.0000 −0.212291 −0.106145 0.994351i \(-0.533851\pi\)
−0.106145 + 0.994351i \(0.533851\pi\)
\(180\) 62.3538i 0.346410i
\(181\) 187.061i 1.03349i −0.856140 0.516744i \(-0.827144\pi\)
0.856140 0.516744i \(-0.172856\pi\)
\(182\) −48.0000 6.92820i −0.263736 0.0380671i
\(183\) −60.0000 −0.327869
\(184\) 98.0000 0.532609
\(185\) 180.133i 0.973693i
\(186\) −48.0000 −0.258065
\(187\) 0 0
\(188\) 83.1384i 0.442226i
\(189\) 36.0000 + 5.19615i 0.190476 + 0.0274929i
\(190\) −144.000 −0.757895
\(191\) 82.0000 0.429319 0.214660 0.976689i \(-0.431136\pi\)
0.214660 + 0.976689i \(0.431136\pi\)
\(192\) 22.5167i 0.117274i
\(193\) 50.0000 0.259067 0.129534 0.991575i \(-0.458652\pi\)
0.129534 + 0.991575i \(0.458652\pi\)
\(194\) 55.4256i 0.285699i
\(195\) 83.1384i 0.426351i
\(196\) 141.000 + 41.5692i 0.719388 + 0.212088i
\(197\) −278.000 −1.41117 −0.705584 0.708627i \(-0.749315\pi\)
−0.705584 + 0.708627i \(0.749315\pi\)
\(198\) −30.0000 −0.151515
\(199\) 290.985i 1.46223i −0.682252 0.731117i \(-0.738999\pi\)
0.682252 0.731117i \(-0.261001\pi\)
\(200\) 161.000 0.805000
\(201\) 128.172i 0.637670i
\(202\) 117.779i 0.583067i
\(203\) −38.0000 + 263.272i −0.187192 + 1.29691i
\(204\) 0 0
\(205\) −480.000 −2.34146
\(206\) 96.9948i 0.470849i
\(207\) 42.0000 0.202899
\(208\) 34.6410i 0.166543i
\(209\) 207.846i 0.994479i
\(210\) 12.0000 83.1384i 0.0571429 0.395897i
\(211\) 74.0000 0.350711 0.175355 0.984505i \(-0.443893\pi\)
0.175355 + 0.984505i \(0.443893\pi\)
\(212\) −30.0000 −0.141509
\(213\) 107.387i 0.504165i
\(214\) 10.0000 0.0467290
\(215\) 180.133i 0.837829i
\(216\) 36.3731i 0.168394i
\(217\) −192.000 27.7128i −0.884793 0.127709i
\(218\) 74.0000 0.339450
\(219\) 72.0000 0.328767
\(220\) 207.846i 0.944755i
\(221\) 0 0
\(222\) 45.0333i 0.202853i
\(223\) 304.841i 1.36700i −0.729951 0.683500i \(-0.760457\pi\)
0.729951 0.683500i \(-0.239543\pi\)
\(224\) 33.0000 228.631i 0.147321 1.02067i
\(225\) 69.0000 0.306667
\(226\) 178.000 0.787611
\(227\) 6.92820i 0.0305207i −0.999884 0.0152604i \(-0.995142\pi\)
0.999884 0.0152604i \(-0.00485771\pi\)
\(228\) −108.000 −0.473684
\(229\) 284.056i 1.24042i 0.784436 + 0.620210i \(0.212953\pi\)
−0.784436 + 0.620210i \(0.787047\pi\)
\(230\) 96.9948i 0.421717i
\(231\) −120.000 17.3205i −0.519481 0.0749806i
\(232\) 266.000 1.14655
\(233\) 178.000 0.763948 0.381974 0.924173i \(-0.375244\pi\)
0.381974 + 0.924173i \(0.375244\pi\)
\(234\) 20.7846i 0.0888231i
\(235\) 192.000 0.817021
\(236\) 228.631i 0.968774i
\(237\) 79.6743i 0.336179i
\(238\) 0 0
\(239\) 34.0000 0.142259 0.0711297 0.997467i \(-0.477340\pi\)
0.0711297 + 0.997467i \(0.477340\pi\)
\(240\) 60.0000 0.250000
\(241\) 193.990i 0.804936i 0.915434 + 0.402468i \(0.131848\pi\)
−0.915434 + 0.402468i \(0.868152\pi\)
\(242\) −21.0000 −0.0867769
\(243\) 15.5885i 0.0641500i
\(244\) 103.923i 0.425914i
\(245\) 96.0000 325.626i 0.391837 1.32908i
\(246\) 120.000 0.487805
\(247\) 144.000 0.582996
\(248\) 193.990i 0.782216i
\(249\) −156.000 −0.626506
\(250\) 13.8564i 0.0554256i
\(251\) 187.061i 0.745265i −0.927979 0.372632i \(-0.878455\pi\)
0.927979 0.372632i \(-0.121545\pi\)
\(252\) 9.00000 62.3538i 0.0357143 0.247436i
\(253\) −140.000 −0.553360
\(254\) −190.000 −0.748031
\(255\) 0 0
\(256\) −171.000 −0.667969
\(257\) 138.564i 0.539160i −0.962978 0.269580i \(-0.913115\pi\)
0.962978 0.269580i \(-0.0868848\pi\)
\(258\) 45.0333i 0.174548i
\(259\) 26.0000 180.133i 0.100386 0.695495i
\(260\) −144.000 −0.553846
\(261\) 114.000 0.436782
\(262\) 48.4974i 0.185105i
\(263\) −62.0000 −0.235741 −0.117871 0.993029i \(-0.537607\pi\)
−0.117871 + 0.993029i \(0.537607\pi\)
\(264\) 121.244i 0.459256i
\(265\) 69.2820i 0.261442i
\(266\) 144.000 + 20.7846i 0.541353 + 0.0781376i
\(267\) 72.0000 0.269663
\(268\) −222.000 −0.828358
\(269\) 228.631i 0.849928i −0.905210 0.424964i \(-0.860287\pi\)
0.905210 0.424964i \(-0.139713\pi\)
\(270\) −36.0000 −0.133333
\(271\) 332.554i 1.22714i 0.789642 + 0.613568i \(0.210266\pi\)
−0.789642 + 0.613568i \(0.789734\pi\)
\(272\) 0 0
\(273\) −12.0000 + 83.1384i −0.0439560 + 0.304536i
\(274\) −206.000 −0.751825
\(275\) −230.000 −0.836364
\(276\) 72.7461i 0.263573i
\(277\) 26.0000 0.0938628 0.0469314 0.998898i \(-0.485056\pi\)
0.0469314 + 0.998898i \(0.485056\pi\)
\(278\) 117.779i 0.423667i
\(279\) 83.1384i 0.297987i
\(280\) −336.000 48.4974i −1.20000 0.173205i
\(281\) −494.000 −1.75801 −0.879004 0.476815i \(-0.841791\pi\)
−0.879004 + 0.476815i \(0.841791\pi\)
\(282\) −48.0000 −0.170213
\(283\) 200.918i 0.709957i 0.934874 + 0.354979i \(0.115512\pi\)
−0.934874 + 0.354979i \(0.884488\pi\)
\(284\) 186.000 0.654930
\(285\) 249.415i 0.875141i
\(286\) 69.2820i 0.242245i
\(287\) 480.000 + 69.2820i 1.67247 + 0.241401i
\(288\) −99.0000 −0.343750
\(289\) 289.000 1.00000
\(290\) 263.272i 0.907834i
\(291\) −96.0000 −0.329897
\(292\) 124.708i 0.427081i
\(293\) 491.902i 1.67885i −0.543477 0.839424i \(-0.682893\pi\)
0.543477 0.839424i \(-0.317107\pi\)
\(294\) −24.0000 + 81.4064i −0.0816327 + 0.276892i
\(295\) 528.000 1.78983
\(296\) −182.000 −0.614865
\(297\) 51.9615i 0.174955i
\(298\) 106.000 0.355705
\(299\) 96.9948i 0.324397i
\(300\) 119.512i 0.398372i
\(301\) 26.0000 180.133i 0.0863787 0.598449i
\(302\) −46.0000 −0.152318
\(303\) 204.000 0.673267
\(304\) 103.923i 0.341852i
\(305\) 240.000 0.786885
\(306\) 0 0
\(307\) 20.7846i 0.0677023i 0.999427 + 0.0338512i \(0.0107772\pi\)
−0.999427 + 0.0338512i \(0.989223\pi\)
\(308\) −30.0000 + 207.846i −0.0974026 + 0.674825i
\(309\) 168.000 0.543689
\(310\) 192.000 0.619355
\(311\) 13.8564i 0.0445544i −0.999752 0.0222772i \(-0.992908\pi\)
0.999752 0.0222772i \(-0.00709163\pi\)
\(312\) 84.0000 0.269231
\(313\) 263.272i 0.841124i −0.907264 0.420562i \(-0.861833\pi\)
0.907264 0.420562i \(-0.138167\pi\)
\(314\) 242.487i 0.772252i
\(315\) −144.000 20.7846i −0.457143 0.0659829i
\(316\) 138.000 0.436709
\(317\) 250.000 0.788644 0.394322 0.918972i \(-0.370980\pi\)
0.394322 + 0.918972i \(0.370980\pi\)
\(318\) 17.3205i 0.0544670i
\(319\) −380.000 −1.19122
\(320\) 90.0666i 0.281458i
\(321\) 17.3205i 0.0539580i
\(322\) −14.0000 + 96.9948i −0.0434783 + 0.301226i
\(323\) 0 0
\(324\) −27.0000 −0.0833333
\(325\) 159.349i 0.490304i
\(326\) 170.000 0.521472
\(327\) 128.172i 0.391963i
\(328\) 484.974i 1.47858i
\(329\) −192.000 27.7128i −0.583587 0.0842335i
\(330\) 120.000 0.363636
\(331\) −646.000 −1.95166 −0.975831 0.218527i \(-0.929875\pi\)
−0.975831 + 0.218527i \(0.929875\pi\)
\(332\) 270.200i 0.813855i
\(333\) −78.0000 −0.234234
\(334\) 96.9948i 0.290404i
\(335\) 512.687i 1.53041i
\(336\) −60.0000 8.66025i −0.178571 0.0257746i
\(337\) 146.000 0.433234 0.216617 0.976257i \(-0.430498\pi\)
0.216617 + 0.976257i \(0.430498\pi\)
\(338\) 121.000 0.357988
\(339\) 308.305i 0.909454i
\(340\) 0 0
\(341\) 277.128i 0.812692i
\(342\) 62.3538i 0.182321i
\(343\) −143.000 + 311.769i −0.416910 + 0.908948i
\(344\) −182.000 −0.529070
\(345\) −168.000 −0.486957
\(346\) 173.205i 0.500593i
\(347\) 106.000 0.305476 0.152738 0.988267i \(-0.451191\pi\)
0.152738 + 0.988267i \(0.451191\pi\)
\(348\) 197.454i 0.567396i
\(349\) 630.466i 1.80649i 0.429121 + 0.903247i \(0.358823\pi\)
−0.429121 + 0.903247i \(0.641177\pi\)
\(350\) −23.0000 + 159.349i −0.0657143 + 0.455282i
\(351\) 36.0000 0.102564
\(352\) 330.000 0.937500
\(353\) 304.841i 0.863572i 0.901976 + 0.431786i \(0.142116\pi\)
−0.901976 + 0.431786i \(0.857884\pi\)
\(354\) −132.000 −0.372881
\(355\) 429.549i 1.21000i
\(356\) 124.708i 0.350302i
\(357\) 0 0
\(358\) −38.0000 −0.106145
\(359\) −494.000 −1.37604 −0.688022 0.725690i \(-0.741521\pi\)
−0.688022 + 0.725690i \(0.741521\pi\)
\(360\) 145.492i 0.404145i
\(361\) −71.0000 −0.196676
\(362\) 187.061i 0.516744i
\(363\) 36.3731i 0.100201i
\(364\) 144.000 + 20.7846i 0.395604 + 0.0571006i
\(365\) −288.000 −0.789041
\(366\) −60.0000 −0.163934
\(367\) 554.256i 1.51024i −0.655589 0.755118i \(-0.727580\pi\)
0.655589 0.755118i \(-0.272420\pi\)
\(368\) −70.0000 −0.190217
\(369\) 207.846i 0.563269i
\(370\) 180.133i 0.486847i
\(371\) 10.0000 69.2820i 0.0269542 0.186744i
\(372\) 144.000 0.387097
\(373\) 218.000 0.584450 0.292225 0.956350i \(-0.405604\pi\)
0.292225 + 0.956350i \(0.405604\pi\)
\(374\) 0 0
\(375\) 24.0000 0.0640000
\(376\) 193.990i 0.515930i
\(377\) 263.272i 0.698333i
\(378\) 36.0000 + 5.19615i 0.0952381 + 0.0137464i
\(379\) −550.000 −1.45119 −0.725594 0.688123i \(-0.758435\pi\)
−0.725594 + 0.688123i \(0.758435\pi\)
\(380\) 432.000 1.13684
\(381\) 329.090i 0.863752i
\(382\) 82.0000 0.214660
\(383\) 55.4256i 0.144714i −0.997379 0.0723572i \(-0.976948\pi\)
0.997379 0.0723572i \(-0.0230522\pi\)
\(384\) 206.114i 0.536755i
\(385\) 480.000 + 69.2820i 1.24675 + 0.179953i
\(386\) 50.0000 0.129534
\(387\) −78.0000 −0.201550
\(388\) 166.277i 0.428549i
\(389\) 394.000 1.01285 0.506427 0.862283i \(-0.330966\pi\)
0.506427 + 0.862283i \(0.330966\pi\)
\(390\) 83.1384i 0.213175i
\(391\) 0 0
\(392\) 329.000 + 96.9948i 0.839286 + 0.247436i
\(393\) 84.0000 0.213740
\(394\) −278.000 −0.705584
\(395\) 318.697i 0.806829i
\(396\) 90.0000 0.227273
\(397\) 103.923i 0.261771i 0.991397 + 0.130885i \(0.0417820\pi\)
−0.991397 + 0.130885i \(0.958218\pi\)
\(398\) 290.985i 0.731117i
\(399\) 36.0000 249.415i 0.0902256 0.625101i
\(400\) −115.000 −0.287500
\(401\) 178.000 0.443890 0.221945 0.975059i \(-0.428759\pi\)
0.221945 + 0.975059i \(0.428759\pi\)
\(402\) 128.172i 0.318835i
\(403\) −192.000 −0.476427
\(404\) 353.338i 0.874600i
\(405\) 62.3538i 0.153960i
\(406\) −38.0000 + 263.272i −0.0935961 + 0.648453i
\(407\) 260.000 0.638821
\(408\) 0 0
\(409\) 651.251i 1.59230i −0.605099 0.796150i \(-0.706866\pi\)
0.605099 0.796150i \(-0.293134\pi\)
\(410\) −480.000 −1.17073
\(411\) 356.802i 0.868133i
\(412\) 290.985i 0.706273i
\(413\) −528.000 76.2102i −1.27845 0.184528i
\(414\) 42.0000 0.101449
\(415\) 624.000 1.50361
\(416\) 228.631i 0.549593i
\(417\) 204.000 0.489209
\(418\) 207.846i 0.497239i
\(419\) 713.605i 1.70311i 0.524262 + 0.851557i \(0.324341\pi\)
−0.524262 + 0.851557i \(0.675659\pi\)
\(420\) −36.0000 + 249.415i −0.0857143 + 0.593846i
\(421\) 602.000 1.42993 0.714964 0.699161i \(-0.246443\pi\)
0.714964 + 0.699161i \(0.246443\pi\)
\(422\) 74.0000 0.175355
\(423\) 83.1384i 0.196545i
\(424\) −70.0000 −0.165094
\(425\) 0 0
\(426\) 107.387i 0.252083i
\(427\) −240.000 34.6410i −0.562061 0.0811265i
\(428\) −30.0000 −0.0700935
\(429\) −120.000 −0.279720
\(430\) 180.133i 0.418915i
\(431\) 34.0000 0.0788863 0.0394432 0.999222i \(-0.487442\pi\)
0.0394432 + 0.999222i \(0.487442\pi\)
\(432\) 25.9808i 0.0601407i
\(433\) 415.692i 0.960028i 0.877261 + 0.480014i \(0.159368\pi\)
−0.877261 + 0.480014i \(0.840632\pi\)
\(434\) −192.000 27.7128i −0.442396 0.0638544i
\(435\) −456.000 −1.04828
\(436\) −222.000 −0.509174
\(437\) 290.985i 0.665869i
\(438\) 72.0000 0.164384
\(439\) 96.9948i 0.220945i −0.993879 0.110472i \(-0.964764\pi\)
0.993879 0.110472i \(-0.0352364\pi\)
\(440\) 484.974i 1.10221i
\(441\) 141.000 + 41.5692i 0.319728 + 0.0942613i
\(442\) 0 0
\(443\) −470.000 −1.06095 −0.530474 0.847701i \(-0.677986\pi\)
−0.530474 + 0.847701i \(0.677986\pi\)
\(444\) 135.100i 0.304279i
\(445\) −288.000 −0.647191
\(446\) 304.841i 0.683500i
\(447\) 183.597i 0.410732i
\(448\) 13.0000 90.0666i 0.0290179 0.201042i
\(449\) 514.000 1.14477 0.572383 0.819986i \(-0.306019\pi\)
0.572383 + 0.819986i \(0.306019\pi\)
\(450\) 69.0000 0.153333
\(451\) 692.820i 1.53619i
\(452\) −534.000 −1.18142
\(453\) 79.6743i 0.175882i
\(454\) 6.92820i 0.0152604i
\(455\) 48.0000 332.554i 0.105495 0.730887i
\(456\) −252.000 −0.552632
\(457\) −478.000 −1.04595 −0.522976 0.852347i \(-0.675178\pi\)
−0.522976 + 0.852347i \(0.675178\pi\)
\(458\) 284.056i 0.620210i
\(459\) 0 0
\(460\) 290.985i 0.632575i
\(461\) 672.036i 1.45778i −0.684632 0.728889i \(-0.740037\pi\)
0.684632 0.728889i \(-0.259963\pi\)
\(462\) −120.000 17.3205i −0.259740 0.0374903i
\(463\) −430.000 −0.928726 −0.464363 0.885645i \(-0.653717\pi\)
−0.464363 + 0.885645i \(0.653717\pi\)
\(464\) −190.000 −0.409483
\(465\) 332.554i 0.715169i
\(466\) 178.000 0.381974
\(467\) 852.169i 1.82477i 0.409330 + 0.912387i \(0.365763\pi\)
−0.409330 + 0.912387i \(0.634237\pi\)
\(468\) 62.3538i 0.133235i
\(469\) 74.0000 512.687i 0.157783 1.09315i
\(470\) 192.000 0.408511
\(471\) 420.000 0.891720
\(472\) 533.472i 1.13024i
\(473\) 260.000 0.549683
\(474\) 79.6743i 0.168089i
\(475\) 478.046i 1.00641i
\(476\) 0 0
\(477\) −30.0000 −0.0628931
\(478\) 34.0000 0.0711297
\(479\) 387.979i 0.809978i 0.914322 + 0.404989i \(0.132725\pi\)
−0.914322 + 0.404989i \(0.867275\pi\)
\(480\) 396.000 0.825000
\(481\) 180.133i 0.374497i
\(482\) 193.990i 0.402468i
\(483\) 168.000 + 24.2487i 0.347826 + 0.0502044i
\(484\) 63.0000 0.130165
\(485\) 384.000 0.791753
\(486\) 15.5885i 0.0320750i
\(487\) 818.000 1.67967 0.839836 0.542841i \(-0.182651\pi\)
0.839836 + 0.542841i \(0.182651\pi\)
\(488\) 242.487i 0.496900i
\(489\) 294.449i 0.602144i
\(490\) 96.0000 325.626i 0.195918 0.664542i
\(491\) 490.000 0.997963 0.498982 0.866613i \(-0.333707\pi\)
0.498982 + 0.866613i \(0.333707\pi\)
\(492\) −360.000 −0.731707
\(493\) 0 0
\(494\) 144.000 0.291498
\(495\) 207.846i 0.419891i
\(496\) 138.564i 0.279363i
\(497\) −62.0000 + 429.549i −0.124748 + 0.864283i
\(498\) −156.000 −0.313253
\(499\) 554.000 1.11022 0.555110 0.831777i \(-0.312676\pi\)
0.555110 + 0.831777i \(0.312676\pi\)
\(500\) 41.5692i 0.0831384i
\(501\) −168.000 −0.335329
\(502\) 187.061i 0.372632i
\(503\) 290.985i 0.578498i −0.957254 0.289249i \(-0.906594\pi\)
0.957254 0.289249i \(-0.0934056\pi\)
\(504\) 21.0000 145.492i 0.0416667 0.288675i
\(505\) −816.000 −1.61584
\(506\) −140.000 −0.276680
\(507\) 209.578i 0.413369i
\(508\) 570.000 1.12205
\(509\) 963.020i 1.89198i 0.324189 + 0.945992i \(0.394909\pi\)
−0.324189 + 0.945992i \(0.605091\pi\)
\(510\) 0 0
\(511\) 288.000 + 41.5692i 0.563601 + 0.0813488i
\(512\) 305.000 0.595703
\(513\) −108.000 −0.210526
\(514\) 138.564i 0.269580i
\(515\) −672.000 −1.30485
\(516\) 135.100i 0.261822i
\(517\) 277.128i 0.536031i
\(518\) 26.0000 180.133i 0.0501931 0.347748i
\(519\) −300.000 −0.578035
\(520\) −336.000 −0.646154
\(521\) 124.708i 0.239362i 0.992812 + 0.119681i \(0.0381872\pi\)
−0.992812 + 0.119681i \(0.961813\pi\)
\(522\) 114.000 0.218391
\(523\) 62.3538i 0.119223i 0.998222 + 0.0596117i \(0.0189862\pi\)
−0.998222 + 0.0596117i \(0.981014\pi\)
\(524\) 145.492i 0.277657i
\(525\) 276.000 + 39.8372i 0.525714 + 0.0758803i
\(526\) −62.0000 −0.117871
\(527\) 0 0
\(528\) 86.6025i 0.164020i
\(529\) −333.000 −0.629490
\(530\) 69.2820i 0.130721i
\(531\) 228.631i 0.430566i
\(532\) −432.000 62.3538i −0.812030 0.117206i
\(533\) 480.000 0.900563
\(534\) 72.0000 0.134831
\(535\) 69.2820i 0.129499i
\(536\) −518.000 −0.966418
\(537\) 65.8179i 0.122566i
\(538\) 228.631i 0.424964i
\(539\) −470.000 138.564i −0.871985 0.257076i
\(540\) 108.000 0.200000
\(541\) −118.000 −0.218115 −0.109057 0.994035i \(-0.534783\pi\)
−0.109057 + 0.994035i \(0.534783\pi\)
\(542\) 332.554i 0.613568i
\(543\) −324.000 −0.596685
\(544\) 0 0
\(545\) 512.687i 0.940710i
\(546\) −12.0000 + 83.1384i −0.0219780 + 0.152268i
\(547\) −406.000 −0.742230 −0.371115 0.928587i \(-0.621024\pi\)
−0.371115 + 0.928587i \(0.621024\pi\)
\(548\) 618.000 1.12774
\(549\) 103.923i 0.189295i
\(550\) −230.000 −0.418182
\(551\) 789.815i 1.43342i
\(552\) 169.741i 0.307502i
\(553\) −46.0000 + 318.697i −0.0831826 + 0.576306i
\(554\) 26.0000 0.0469314
\(555\) 312.000 0.562162
\(556\) 353.338i 0.635501i
\(557\) −998.000 −1.79174 −0.895871 0.444315i \(-0.853447\pi\)
−0.895871 + 0.444315i \(0.853447\pi\)
\(558\) 83.1384i 0.148994i
\(559\) 180.133i 0.322242i
\(560\) 240.000 + 34.6410i 0.428571 + 0.0618590i
\(561\) 0 0
\(562\) −494.000 −0.879004
\(563\) 367.195i 0.652211i −0.945333 0.326105i \(-0.894264\pi\)
0.945333 0.326105i \(-0.105736\pi\)
\(564\) 144.000 0.255319
\(565\) 1233.22i 2.18269i
\(566\) 200.918i 0.354979i
\(567\) 9.00000 62.3538i 0.0158730 0.109971i
\(568\) 434.000 0.764085
\(569\) −62.0000 −0.108963 −0.0544815 0.998515i \(-0.517351\pi\)
−0.0544815 + 0.998515i \(0.517351\pi\)
\(570\) 249.415i 0.437571i
\(571\) 410.000 0.718039 0.359019 0.933330i \(-0.383111\pi\)
0.359019 + 0.933330i \(0.383111\pi\)
\(572\) 207.846i 0.363367i
\(573\) 142.028i 0.247868i
\(574\) 480.000 + 69.2820i 0.836237 + 0.120700i
\(575\) 322.000 0.560000
\(576\) −39.0000 −0.0677083
\(577\) 831.384i 1.44087i −0.693520 0.720437i \(-0.743941\pi\)
0.693520 0.720437i \(-0.256059\pi\)
\(578\) 289.000 0.500000
\(579\) 86.6025i 0.149573i
\(580\) 789.815i 1.36175i
\(581\) −624.000 90.0666i −1.07401 0.155020i
\(582\) −96.0000 −0.164948
\(583\) 100.000 0.171527
\(584\) 290.985i 0.498261i
\(585\) −144.000 −0.246154
\(586\) 491.902i 0.839424i
\(587\) 547.328i 0.932416i −0.884675 0.466208i \(-0.845620\pi\)
0.884675 0.466208i \(-0.154380\pi\)
\(588\) 72.0000 244.219i 0.122449 0.415339i
\(589\) 576.000 0.977929
\(590\) 528.000 0.894915
\(591\) 481.510i 0.814738i
\(592\) 130.000 0.219595
\(593\) 997.661i 1.68240i −0.540727 0.841198i \(-0.681851\pi\)
0.540727 0.841198i \(-0.318149\pi\)
\(594\) 51.9615i 0.0874773i
\(595\) 0 0
\(596\) −318.000 −0.533557
\(597\) −504.000 −0.844221
\(598\) 96.9948i 0.162199i
\(599\) 1186.00 1.97997 0.989983 0.141184i \(-0.0450911\pi\)
0.989983 + 0.141184i \(0.0450911\pi\)
\(600\) 278.860i 0.464767i
\(601\) 401.836i 0.668612i 0.942465 + 0.334306i \(0.108502\pi\)
−0.942465 + 0.334306i \(0.891498\pi\)
\(602\) 26.0000 180.133i 0.0431894 0.299225i
\(603\) −222.000 −0.368159
\(604\) 138.000 0.228477
\(605\) 145.492i 0.240483i
\(606\) 204.000 0.336634
\(607\) 138.564i 0.228277i 0.993465 + 0.114138i \(0.0364107\pi\)
−0.993465 + 0.114138i \(0.963589\pi\)
\(608\) 685.892i 1.12811i
\(609\) 456.000 + 65.8179i 0.748768 + 0.108075i
\(610\) 240.000 0.393443
\(611\) −192.000 −0.314239
\(612\) 0 0
\(613\) 26.0000 0.0424144 0.0212072 0.999775i \(-0.493249\pi\)
0.0212072 + 0.999775i \(0.493249\pi\)
\(614\) 20.7846i 0.0338512i
\(615\) 831.384i 1.35184i
\(616\) −70.0000 + 484.974i −0.113636 + 0.787296i
\(617\) −734.000 −1.18963 −0.594814 0.803864i \(-0.702774\pi\)
−0.594814 + 0.803864i \(0.702774\pi\)
\(618\) 168.000 0.271845
\(619\) 838.313i 1.35430i 0.735844 + 0.677151i \(0.236785\pi\)
−0.735844 + 0.677151i \(0.763215\pi\)
\(620\) −576.000 −0.929032
\(621\) 72.7461i 0.117144i
\(622\) 13.8564i 0.0222772i
\(623\) 288.000 + 41.5692i 0.462279 + 0.0667243i
\(624\) −60.0000 −0.0961538
\(625\) −671.000 −1.07360
\(626\) 263.272i 0.420562i
\(627\) 360.000 0.574163
\(628\) 727.461i 1.15838i
\(629\) 0 0
\(630\) −144.000 20.7846i −0.228571 0.0329914i
\(631\) −286.000 −0.453249 −0.226624 0.973982i \(-0.572769\pi\)
−0.226624 + 0.973982i \(0.572769\pi\)
\(632\) 322.000 0.509494
\(633\) 128.172i 0.202483i
\(634\) 250.000 0.394322
\(635\) 1316.36i 2.07301i
\(636\) 51.9615i 0.0817005i
\(637\) −96.0000 + 325.626i −0.150706 + 0.511186i
\(638\) −380.000 −0.595611
\(639\) 186.000 0.291080
\(640\) 824.456i 1.28821i
\(641\) 658.000 1.02652 0.513261 0.858233i \(-0.328437\pi\)
0.513261 + 0.858233i \(0.328437\pi\)
\(642\) 17.3205i 0.0269790i
\(643\) 325.626i 0.506416i 0.967412 + 0.253208i \(0.0814857\pi\)
−0.967412 + 0.253208i \(0.918514\pi\)
\(644\) 42.0000 290.985i 0.0652174 0.451839i
\(645\) 312.000 0.483721
\(646\) 0 0
\(647\) 457.261i 0.706741i 0.935483 + 0.353370i \(0.114964\pi\)
−0.935483 + 0.353370i \(0.885036\pi\)
\(648\) −63.0000 −0.0972222
\(649\) 762.102i 1.17427i
\(650\) 159.349i 0.245152i
\(651\) −48.0000 + 332.554i −0.0737327 + 0.510835i
\(652\) −510.000 −0.782209
\(653\) −614.000 −0.940276 −0.470138 0.882593i \(-0.655796\pi\)
−0.470138 + 0.882593i \(0.655796\pi\)
\(654\) 128.172i 0.195981i
\(655\) −336.000 −0.512977
\(656\) 346.410i 0.528064i
\(657\) 124.708i 0.189814i
\(658\) −192.000 27.7128i −0.291793 0.0421167i
\(659\) 442.000 0.670713 0.335357 0.942091i \(-0.391143\pi\)
0.335357 + 0.942091i \(0.391143\pi\)
\(660\) −360.000 −0.545455
\(661\) 34.6410i 0.0524070i 0.999657 + 0.0262035i \(0.00834179\pi\)
−0.999657 + 0.0262035i \(0.991658\pi\)
\(662\) −646.000 −0.975831
\(663\) 0 0
\(664\) 630.466i 0.949498i
\(665\) −144.000 + 997.661i −0.216541 + 1.50024i
\(666\) −78.0000 −0.117117
\(667\) 532.000 0.797601
\(668\) 290.985i 0.435606i
\(669\) −528.000 −0.789238
\(670\) 512.687i 0.765205i
\(671\) 346.410i 0.516260i
\(672\) −396.000 57.1577i −0.589286 0.0850561i
\(673\) 386.000 0.573551 0.286776 0.957998i \(-0.407417\pi\)
0.286776 + 0.957998i \(0.407417\pi\)
\(674\) 146.000 0.216617
\(675\) 119.512i 0.177054i
\(676\) −363.000 −0.536982
\(677\) 782.887i 1.15641i 0.815893 + 0.578203i \(0.196246\pi\)
−0.815893 + 0.578203i \(0.803754\pi\)
\(678\) 308.305i 0.454727i
\(679\) −384.000 55.4256i −0.565538 0.0816283i
\(680\) 0 0
\(681\) −12.0000 −0.0176211
\(682\) 277.128i 0.406346i
\(683\) 298.000 0.436310 0.218155 0.975914i \(-0.429996\pi\)
0.218155 + 0.975914i \(0.429996\pi\)
\(684\) 187.061i 0.273482i
\(685\) 1427.21i 2.08352i
\(686\) −143.000 + 311.769i −0.208455 + 0.454474i
\(687\) 492.000 0.716157
\(688\) 130.000 0.188953
\(689\) 69.2820i 0.100554i
\(690\) −168.000 −0.243478
\(691\) 866.025i 1.25329i −0.779304 0.626646i \(-0.784427\pi\)
0.779304 0.626646i \(-0.215573\pi\)
\(692\) 519.615i 0.750889i
\(693\) −30.0000 + 207.846i −0.0432900 + 0.299922i
\(694\) 106.000 0.152738
\(695\) −816.000 −1.17410
\(696\) 460.726i 0.661962i
\(697\) 0 0
\(698\) 630.466i 0.903247i
\(699\) 308.305i 0.441066i
\(700\) 69.0000 478.046i 0.0985714 0.682923i
\(701\) 154.000 0.219686 0.109843 0.993949i \(-0.464965\pi\)
0.109843 + 0.993949i \(0.464965\pi\)
\(702\) 36.0000 0.0512821
\(703\) 540.400i 0.768705i
\(704\) 130.000 0.184659
\(705\) 332.554i 0.471707i
\(706\) 304.841i 0.431786i
\(707\) 816.000 + 117.779i 1.15417 + 0.166590i
\(708\) 396.000 0.559322
\(709\) 890.000 1.25529 0.627645 0.778500i \(-0.284019\pi\)
0.627645 + 0.778500i \(0.284019\pi\)
\(710\) 429.549i 0.604998i
\(711\) 138.000 0.194093
\(712\) 290.985i 0.408686i
\(713\) 387.979i 0.544151i
\(714\) 0 0
\(715\) 480.000 0.671329
\(716\) 114.000 0.159218
\(717\) 58.8897i 0.0821335i
\(718\) −494.000 −0.688022
\(719\) 83.1384i 0.115631i −0.998327 0.0578153i \(-0.981587\pi\)
0.998327 0.0578153i \(-0.0184135\pi\)
\(720\) 103.923i 0.144338i
\(721\) 672.000 + 96.9948i 0.932039 + 0.134528i
\(722\) −71.0000 −0.0983380
\(723\) 336.000 0.464730
\(724\) 561.184i 0.775117i
\(725\) 874.000 1.20552
\(726\) 36.3731i 0.0501006i
\(727\) 235.559i 0.324015i 0.986790 + 0.162008i \(0.0517969\pi\)
−0.986790 + 0.162008i \(0.948203\pi\)
\(728\) 336.000 + 48.4974i 0.461538 + 0.0666173i
\(729\) −27.0000 −0.0370370
\(730\) −288.000 −0.394521
\(731\) 0 0
\(732\) 180.000 0.245902
\(733\) 76.2102i 0.103970i 0.998648 + 0.0519852i \(0.0165549\pi\)
−0.998648 + 0.0519852i \(0.983445\pi\)
\(734\) 554.256i 0.755118i
\(735\) −564.000 166.277i −0.767347 0.226227i
\(736\) −462.000 −0.627717
\(737\) 740.000 1.00407
\(738\) 207.846i 0.281634i
\(739\) −982.000 −1.32882 −0.664411 0.747367i \(-0.731318\pi\)
−0.664411 + 0.747367i \(0.731318\pi\)
\(740\) 540.400i 0.730270i
\(741\) 249.415i 0.336593i
\(742\) 10.0000 69.2820i 0.0134771 0.0933720i
\(743\) −686.000 −0.923284 −0.461642 0.887066i \(-0.652740\pi\)
−0.461642 + 0.887066i \(0.652740\pi\)
\(744\) 336.000 0.451613
\(745\) 734.390i 0.985758i
\(746\) 218.000 0.292225
\(747\) 270.200i 0.361713i
\(748\) 0 0
\(749\) 10.0000 69.2820i 0.0133511 0.0924994i
\(750\) 24.0000 0.0320000
\(751\) 290.000 0.386152 0.193076 0.981184i \(-0.438154\pi\)
0.193076 + 0.981184i \(0.438154\pi\)
\(752\) 138.564i 0.184261i
\(753\) −324.000 −0.430279
\(754\) 263.272i 0.349167i
\(755\) 318.697i 0.422116i
\(756\) −108.000 15.5885i −0.142857 0.0206197i
\(757\) −358.000 −0.472919 −0.236460 0.971641i \(-0.575987\pi\)
−0.236460 + 0.971641i \(0.575987\pi\)
\(758\) −550.000 −0.725594
\(759\) 242.487i 0.319482i
\(760\) 1008.00 1.32632
\(761\) 651.251i 0.855783i 0.903830 + 0.427892i \(0.140744\pi\)
−0.903830 + 0.427892i \(0.859256\pi\)
\(762\) 329.090i 0.431876i
\(763\) 74.0000 512.687i 0.0969856 0.671936i
\(764\) −246.000 −0.321990
\(765\) 0 0
\(766\) 55.4256i 0.0723572i
\(767\) −528.000 −0.688396
\(768\) 296.181i 0.385652i
\(769\) 775.959i 1.00905i −0.863397 0.504525i \(-0.831668\pi\)
0.863397 0.504525i \(-0.168332\pi\)
\(770\) 480.000 + 69.2820i 0.623377 + 0.0899767i
\(771\) −240.000 −0.311284
\(772\) −150.000 −0.194301
\(773\) 228.631i 0.295771i 0.989005 + 0.147885i \(0.0472467\pi\)
−0.989005 + 0.147885i \(0.952753\pi\)
\(774\) −78.0000 −0.100775
\(775\) 637.395i 0.822445i
\(776\) 387.979i 0.499973i
\(777\) −312.000 45.0333i −0.401544 0.0579579i
\(778\) 394.000 0.506427
\(779\) −1440.00 −1.84852
\(780\) 249.415i 0.319763i
\(781\) −620.000 −0.793854
\(782\) 0 0
\(783\) 197.454i 0.252176i
\(784\) −235.000 69.2820i −0.299745 0.0883699i
\(785\) −1680.00 −2.14013
\(786\) 84.0000 0.106870
\(787\) 76.2102i 0.0968364i 0.998827 + 0.0484182i \(0.0154180\pi\)
−0.998827 + 0.0484182i \(0.984582\pi\)
\(788\) 834.000 1.05838
\(789\) 107.387i 0.136105i
\(790\) 318.697i 0.403414i
\(791\) 178.000 1233.22i 0.225032 1.55906i
\(792\) 210.000 0.265152
\(793\) −240.000 −0.302648
\(794\) 103.923i 0.130885i
\(795\) 120.000 0.150943
\(796\) 872.954i 1.09668i
\(797\) 866.025i 1.08661i −0.839537 0.543303i \(-0.817173\pi\)
0.839537 0.543303i \(-0.182827\pi\)
\(798\) 36.0000 249.415i 0.0451128 0.312551i
\(799\) 0 0
\(800\) −759.000 −0.948750
\(801\) 124.708i 0.155690i
\(802\) 178.000 0.221945
\(803\) 415.692i 0.517674i
\(804\) 384.515i 0.478253i
\(805\) −672.000 96.9948i −0.834783 0.120490i
\(806\) −192.000 −0.238213
\(807\) −396.000 −0.490706
\(808\) 824.456i 1.02037i
\(809\) 802.000 0.991347 0.495674 0.868509i \(-0.334921\pi\)
0.495674 + 0.868509i \(0.334921\pi\)
\(810\) 62.3538i 0.0769800i
\(811\) 893.738i 1.10202i 0.834499 + 0.551010i \(0.185757\pi\)
−0.834499 + 0.551010i \(0.814243\pi\)
\(812\) 114.000 789.815i 0.140394 0.972679i
\(813\) 576.000 0.708487
\(814\) 260.000 0.319410
\(815\) 1177.79i 1.44515i
\(816\) 0 0
\(817\) 540.400i 0.661444i
\(818\) 651.251i 0.796150i
\(819\) 144.000 + 20.7846i 0.175824 + 0.0253780i
\(820\) 1440.00 1.75610
\(821\) 1450.00 1.76614 0.883069 0.469242i \(-0.155473\pi\)
0.883069 + 0.469242i \(0.155473\pi\)
\(822\) 356.802i 0.434066i
\(823\) −1246.00 −1.51397 −0.756987 0.653430i \(-0.773329\pi\)
−0.756987 + 0.653430i \(0.773329\pi\)
\(824\) 678.964i 0.823985i
\(825\) 398.372i 0.482875i
\(826\) −528.000 76.2102i −0.639225 0.0922642i
\(827\) 586.000 0.708585 0.354293 0.935135i \(-0.384722\pi\)
0.354293 + 0.935135i \(0.384722\pi\)
\(828\) −126.000 −0.152174
\(829\) 810.600i 0.977804i −0.872339 0.488902i \(-0.837398\pi\)
0.872339 0.488902i \(-0.162602\pi\)
\(830\) 624.000 0.751807
\(831\) 45.0333i 0.0541917i
\(832\) 90.0666i 0.108253i
\(833\) 0 0
\(834\) 204.000 0.244604
\(835\) 672.000 0.804790
\(836\) 623.538i 0.745859i
\(837\) 144.000 0.172043
\(838\) 713.605i 0.851557i
\(839\) 13.8564i 0.0165154i 0.999966 + 0.00825769i \(0.00262853\pi\)
−0.999966 + 0.00825769i \(0.997371\pi\)
\(840\) −84.0000 + 581.969i −0.100000 + 0.692820i
\(841\) 603.000 0.717004
\(842\) 602.000 0.714964
\(843\) 855.633i 1.01499i
\(844\) −222.000 −0.263033
\(845\) 838.313i 0.992086i
\(846\) 83.1384i 0.0982724i
\(847\) −21.0000 + 145.492i −0.0247934 + 0.171774i
\(848\) 50.0000 0.0589623
\(849\) 348.000 0.409894
\(850\) 0 0
\(851\) −364.000 −0.427732
\(852\) 322.161i 0.378124i
\(853\) 242.487i 0.284276i −0.989847 0.142138i \(-0.954602\pi\)
0.989847 0.142138i \(-0.0453976\pi\)
\(854\) −240.000 34.6410i −0.281030 0.0405633i
\(855\) 432.000 0.505263
\(856\) −70.0000 −0.0817757
\(857\) 1399.50i 1.63302i −0.577332 0.816509i \(-0.695906\pi\)
0.577332 0.816509i \(-0.304094\pi\)
\(858\) −120.000 −0.139860
\(859\) 297.913i 0.346813i −0.984850 0.173407i \(-0.944522\pi\)
0.984850 0.173407i \(-0.0554775\pi\)
\(860\) 540.400i 0.628372i
\(861\) 120.000 831.384i 0.139373 0.965603i
\(862\) 34.0000 0.0394432
\(863\) −1070.00 −1.23986 −0.619930 0.784657i \(-0.712839\pi\)
−0.619930 + 0.784657i \(0.712839\pi\)
\(864\) 171.473i 0.198464i
\(865\) 1200.00 1.38728
\(866\) 415.692i 0.480014i
\(867\) 500.563i 0.577350i
\(868\) 576.000 + 83.1384i 0.663594 + 0.0957816i
\(869\) −460.000 −0.529344
\(870\) −456.000 −0.524138
\(871\) 512.687i 0.588619i
\(872\) −518.000 −0.594037
\(873\) 166.277i 0.190466i
\(874\) 290.985i 0.332934i
\(875\) 96.0000 + 13.8564i 0.109714 + 0.0158359i
\(876\) −216.000 −0.246575
\(877\) −502.000 −0.572406 −0.286203 0.958169i \(-0.592393\pi\)
−0.286203 + 0.958169i \(0.592393\pi\)
\(878\) 96.9948i 0.110472i
\(879\) −852.000 −0.969283
\(880\) 346.410i 0.393648i
\(881\) 997.661i 1.13242i 0.824261 + 0.566210i \(0.191591\pi\)
−0.824261 + 0.566210i \(0.808409\pi\)
\(882\) 141.000 + 41.5692i 0.159864 + 0.0471306i
\(883\) −118.000 −0.133635 −0.0668177 0.997765i \(-0.521285\pi\)
−0.0668177 + 0.997765i \(0.521285\pi\)
\(884\) 0 0
\(885\) 914.523i 1.03336i
\(886\) −470.000 −0.530474
\(887\) 512.687i 0.578001i −0.957329 0.289001i \(-0.906677\pi\)
0.957329 0.289001i \(-0.0933230\pi\)
\(888\) 315.233i 0.354992i
\(889\) −190.000 + 1316.36i −0.213723 + 1.48072i
\(890\) −288.000 −0.323596
\(891\) 90.0000 0.101010
\(892\) 914.523i 1.02525i
\(893\) 576.000 0.645017
\(894\) 183.597i 0.205366i
\(895\) 263.272i 0.294158i
\(896\) −119.000 + 824.456i −0.132812 + 0.920152i
\(897\) 168.000 0.187291
\(898\) 514.000 0.572383
\(899\) 1053.09i 1.17140i
\(900\) −207.000 −0.230000
\(901\) 0 0
\(902\) 692.820i 0.768093i
\(903\) −312.000 45.0333i −0.345515 0.0498708i
\(904\) −1246.00 −1.37832
\(905\) 1296.00 1.43204
\(906\) 79.6743i 0.0879408i
\(907\) 1466.00 1.61632 0.808159 0.588965i \(-0.200464\pi\)
0.808159 + 0.588965i \(0.200464\pi\)
\(908\) 20.7846i 0.0228905i
\(909\) 353.338i 0.388711i
\(910\) 48.0000 332.554i 0.0527473 0.365444i
\(911\) −542.000 −0.594951 −0.297475 0.954730i \(-0.596145\pi\)
−0.297475 + 0.954730i \(0.596145\pi\)
\(912\) 180.000 0.197368
\(913\) 900.666i 0.986491i
\(914\) −478.000 −0.522976
\(915\) 415.692i 0.454308i
\(916\) 852.169i 0.930315i
\(917\) 336.000 + 48.4974i 0.366412 + 0.0528870i
\(918\) 0 0
\(919\) −1630.00 −1.77367 −0.886834 0.462089i \(-0.847100\pi\)
−0.886834 + 0.462089i \(0.847100\pi\)
\(920\) 678.964i 0.738004i
\(921\) 36.0000 0.0390879
\(922\) 672.036i 0.728889i
\(923\) 429.549i 0.465383i
\(924\) 360.000 + 51.9615i 0.389610 + 0.0562354i
\(925\) −598.000 −0.646486
\(926\) −430.000 −0.464363
\(927\) 290.985i 0.313899i
\(928\) −1254.00 −1.35129
\(929\) 193.990i 0.208816i −0.994535 0.104408i \(-0.966705\pi\)
0.994535 0.104408i \(-0.0332947\pi\)
\(930\) 332.554i 0.357585i
\(931\) 288.000 976.877i 0.309345 1.04928i
\(932\) −534.000 −0.572961
\(933\) −24.0000 −0.0257235
\(934\) 852.169i 0.912387i
\(935\) 0 0
\(936\) 145.492i 0.155440i
\(937\) 1205.51i 1.28656i 0.765631 + 0.643280i \(0.222427\pi\)
−0.765631 + 0.643280i \(0.777573\pi\)
\(938\) 74.0000 512.687i 0.0788913 0.546575i
\(939\) −456.000 −0.485623
\(940\) −576.000 −0.612766
\(941\) 879.882i 0.935050i 0.883980 + 0.467525i \(0.154854\pi\)
−0.883980 + 0.467525i \(0.845146\pi\)
\(942\) 420.000 0.445860
\(943\) 969.948i 1.02858i
\(944\) 381.051i 0.403656i
\(945\) −36.0000 + 249.415i −0.0380952 + 0.263932i
\(946\) 260.000 0.274841
\(947\) 1402.00 1.48046 0.740232 0.672351i \(-0.234716\pi\)
0.740232 + 0.672351i \(0.234716\pi\)
\(948\) 239.023i 0.252134i
\(949\) 288.000 0.303477
\(950\) 478.046i 0.503206i
\(951\) 433.013i 0.455324i
\(952\) 0 0
\(953\) −110.000 −0.115425 −0.0577125 0.998333i \(-0.518381\pi\)
−0.0577125 + 0.998333i \(0.518381\pi\)
\(954\) −30.0000 −0.0314465
\(955\) 568.113i 0.594882i
\(956\) −102.000 −0.106695
\(957\) 658.179i 0.687753i
\(958\) 387.979i 0.404989i
\(959\) −206.000 + 1427.21i −0.214807 + 1.48823i
\(960\) 156.000 0.162500
\(961\) 193.000 0.200832
\(962\) 180.133i 0.187249i
\(963\) −30.0000 −0.0311526
\(964\) 581.969i 0.603702i
\(965\) 346.410i 0.358974i
\(966\) 168.000 + 24.2487i 0.173913 + 0.0251022i
\(967\) 1202.00 1.24302 0.621510 0.783406i \(-0.286520\pi\)
0.621510 + 0.783406i \(0.286520\pi\)
\(968\) 147.000 0.151860
\(969\) 0 0
\(970\) 384.000 0.395876
\(971\) 270.200i 0.278270i −0.990273 0.139135i \(-0.955568\pi\)
0.990273 0.139135i \(-0.0444322\pi\)
\(972\) 46.7654i 0.0481125i
\(973\) 816.000 + 117.779i 0.838643 + 0.121048i
\(974\) 818.000 0.839836
\(975\) 276.000 0.283077
\(976\) 173.205i 0.177464i
\(977\) −1310.00 −1.34084 −0.670420 0.741982i \(-0.733886\pi\)
−0.670420 + 0.741982i \(0.733886\pi\)
\(978\) 294.449i 0.301072i
\(979\) 415.692i 0.424609i
\(980\) −288.000 + 976.877i −0.293878 + 0.996813i
\(981\) −222.000 −0.226300
\(982\) 490.000 0.498982
\(983\) 1565.77i 1.59285i −0.604736 0.796426i \(-0.706721\pi\)
0.604736 0.796426i \(-0.293279\pi\)
\(984\) −840.000 −0.853659
\(985\) 1926.04i 1.95537i
\(986\) 0 0
\(987\) −48.0000 + 332.554i −0.0486322 + 0.336934i
\(988\) −432.000 −0.437247
\(989\) −364.000 −0.368049
\(990\) 207.846i 0.209946i
\(991\) 818.000 0.825429 0.412714 0.910860i \(-0.364581\pi\)
0.412714 + 0.910860i \(0.364581\pi\)
\(992\) 914.523i 0.921898i
\(993\) 1118.90i 1.12679i
\(994\) −62.0000 + 429.549i −0.0623742 + 0.432141i
\(995\) 2016.00 2.02613
\(996\) 468.000 0.469880
\(997\) 173.205i 0.173726i 0.996220 + 0.0868631i \(0.0276843\pi\)
−0.996220 + 0.0868631i \(0.972316\pi\)
\(998\) 554.000 0.555110
\(999\) 135.100i 0.135235i
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 21.3.d.a.13.1 2
3.2 odd 2 63.3.d.b.55.1 2
4.3 odd 2 336.3.f.a.97.2 2
5.2 odd 4 525.3.e.a.349.3 4
5.3 odd 4 525.3.e.a.349.2 4
5.4 even 2 525.3.h.a.76.2 2
7.2 even 3 147.3.f.d.31.1 2
7.3 odd 6 147.3.f.d.19.1 2
7.4 even 3 147.3.f.b.19.1 2
7.5 odd 6 147.3.f.b.31.1 2
7.6 odd 2 inner 21.3.d.a.13.2 yes 2
8.3 odd 2 1344.3.f.b.769.1 2
8.5 even 2 1344.3.f.c.769.2 2
12.11 even 2 1008.3.f.d.433.1 2
21.2 odd 6 441.3.m.d.325.1 2
21.5 even 6 441.3.m.f.325.1 2
21.11 odd 6 441.3.m.f.19.1 2
21.17 even 6 441.3.m.d.19.1 2
21.20 even 2 63.3.d.b.55.2 2
28.27 even 2 336.3.f.a.97.1 2
35.13 even 4 525.3.e.a.349.1 4
35.27 even 4 525.3.e.a.349.4 4
35.34 odd 2 525.3.h.a.76.1 2
56.13 odd 2 1344.3.f.c.769.1 2
56.27 even 2 1344.3.f.b.769.2 2
84.83 odd 2 1008.3.f.d.433.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.3.d.a.13.1 2 1.1 even 1 trivial
21.3.d.a.13.2 yes 2 7.6 odd 2 inner
63.3.d.b.55.1 2 3.2 odd 2
63.3.d.b.55.2 2 21.20 even 2
147.3.f.b.19.1 2 7.4 even 3
147.3.f.b.31.1 2 7.5 odd 6
147.3.f.d.19.1 2 7.3 odd 6
147.3.f.d.31.1 2 7.2 even 3
336.3.f.a.97.1 2 28.27 even 2
336.3.f.a.97.2 2 4.3 odd 2
441.3.m.d.19.1 2 21.17 even 6
441.3.m.d.325.1 2 21.2 odd 6
441.3.m.f.19.1 2 21.11 odd 6
441.3.m.f.325.1 2 21.5 even 6
525.3.e.a.349.1 4 35.13 even 4
525.3.e.a.349.2 4 5.3 odd 4
525.3.e.a.349.3 4 5.2 odd 4
525.3.e.a.349.4 4 35.27 even 4
525.3.h.a.76.1 2 35.34 odd 2
525.3.h.a.76.2 2 5.4 even 2
1008.3.f.d.433.1 2 12.11 even 2
1008.3.f.d.433.2 2 84.83 odd 2
1344.3.f.b.769.1 2 8.3 odd 2
1344.3.f.b.769.2 2 56.27 even 2
1344.3.f.c.769.1 2 56.13 odd 2
1344.3.f.c.769.2 2 8.5 even 2