Properties

Label 90.3.g.b.37.1
Level $90$
Weight $3$
Character 90.37
Analytic conductor $2.452$
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] = [90,3,Mod(37,90)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(90, base_ring=CyclotomicField(4))
 
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("90.37");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 90.g (of order \(4\), degree \(2\), 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: \(2.45232237924\)
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: \( 1 \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 37.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 90.37
Dual form 90.3.g.b.73.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.00000 + 1.00000i) q^{2} +2.00000i q^{4} +5.00000i q^{5} +(2.00000 + 2.00000i) q^{7} +(-2.00000 + 2.00000i) q^{8} +O(q^{10})\) \(q+(1.00000 + 1.00000i) q^{2} +2.00000i q^{4} +5.00000i q^{5} +(2.00000 + 2.00000i) q^{7} +(-2.00000 + 2.00000i) q^{8} +(-5.00000 + 5.00000i) q^{10} +8.00000 q^{11} +(3.00000 - 3.00000i) q^{13} +4.00000i q^{14} -4.00000 q^{16} +(-7.00000 - 7.00000i) q^{17} -20.0000i q^{19} -10.0000 q^{20} +(8.00000 + 8.00000i) q^{22} +(2.00000 - 2.00000i) q^{23} -25.0000 q^{25} +6.00000 q^{26} +(-4.00000 + 4.00000i) q^{28} -40.0000i q^{29} +52.0000 q^{31} +(-4.00000 - 4.00000i) q^{32} -14.0000i q^{34} +(-10.0000 + 10.0000i) q^{35} +(-3.00000 - 3.00000i) q^{37} +(20.0000 - 20.0000i) q^{38} +(-10.0000 - 10.0000i) q^{40} +8.00000 q^{41} +(-42.0000 + 42.0000i) q^{43} +16.0000i q^{44} +4.00000 q^{46} +(18.0000 + 18.0000i) q^{47} -41.0000i q^{49} +(-25.0000 - 25.0000i) q^{50} +(6.00000 + 6.00000i) q^{52} +(-53.0000 + 53.0000i) q^{53} +40.0000i q^{55} -8.00000 q^{56} +(40.0000 - 40.0000i) q^{58} +20.0000i q^{59} -48.0000 q^{61} +(52.0000 + 52.0000i) q^{62} -8.00000i q^{64} +(15.0000 + 15.0000i) q^{65} +(62.0000 + 62.0000i) q^{67} +(14.0000 - 14.0000i) q^{68} -20.0000 q^{70} +28.0000 q^{71} +(-47.0000 + 47.0000i) q^{73} -6.00000i q^{74} +40.0000 q^{76} +(16.0000 + 16.0000i) q^{77} -20.0000i q^{80} +(8.00000 + 8.00000i) q^{82} +(-18.0000 + 18.0000i) q^{83} +(35.0000 - 35.0000i) q^{85} -84.0000 q^{86} +(-16.0000 + 16.0000i) q^{88} -80.0000i q^{89} +12.0000 q^{91} +(4.00000 + 4.00000i) q^{92} +36.0000i q^{94} +100.000 q^{95} +(-63.0000 - 63.0000i) q^{97} +(41.0000 - 41.0000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 4 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 4 q^{7} - 4 q^{8} - 10 q^{10} + 16 q^{11} + 6 q^{13} - 8 q^{16} - 14 q^{17} - 20 q^{20} + 16 q^{22} + 4 q^{23} - 50 q^{25} + 12 q^{26} - 8 q^{28} + 104 q^{31} - 8 q^{32} - 20 q^{35} - 6 q^{37} + 40 q^{38} - 20 q^{40} + 16 q^{41} - 84 q^{43} + 8 q^{46} + 36 q^{47} - 50 q^{50} + 12 q^{52} - 106 q^{53} - 16 q^{56} + 80 q^{58} - 96 q^{61} + 104 q^{62} + 30 q^{65} + 124 q^{67} + 28 q^{68} - 40 q^{70} + 56 q^{71} - 94 q^{73} + 80 q^{76} + 32 q^{77} + 16 q^{82} - 36 q^{83} + 70 q^{85} - 168 q^{86} - 32 q^{88} + 24 q^{91} + 8 q^{92} + 200 q^{95} - 126 q^{97} + 82 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\)

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 + 1.00000i 0.500000 + 0.500000i
\(3\) 0 0
\(4\) 2.00000i 0.500000i
\(5\) 5.00000i 1.00000i
\(6\) 0 0
\(7\) 2.00000 + 2.00000i 0.285714 + 0.285714i 0.835383 0.549669i \(-0.185246\pi\)
−0.549669 + 0.835383i \(0.685246\pi\)
\(8\) −2.00000 + 2.00000i −0.250000 + 0.250000i
\(9\) 0 0
\(10\) −5.00000 + 5.00000i −0.500000 + 0.500000i
\(11\) 8.00000 0.727273 0.363636 0.931541i \(-0.381535\pi\)
0.363636 + 0.931541i \(0.381535\pi\)
\(12\) 0 0
\(13\) 3.00000 3.00000i 0.230769 0.230769i −0.582245 0.813014i \(-0.697825\pi\)
0.813014 + 0.582245i \(0.197825\pi\)
\(14\) 4.00000i 0.285714i
\(15\) 0 0
\(16\) −4.00000 −0.250000
\(17\) −7.00000 7.00000i −0.411765 0.411765i 0.470588 0.882353i \(-0.344042\pi\)
−0.882353 + 0.470588i \(0.844042\pi\)
\(18\) 0 0
\(19\) 20.0000i 1.05263i −0.850289 0.526316i \(-0.823573\pi\)
0.850289 0.526316i \(-0.176427\pi\)
\(20\) −10.0000 −0.500000
\(21\) 0 0
\(22\) 8.00000 + 8.00000i 0.363636 + 0.363636i
\(23\) 2.00000 2.00000i 0.0869565 0.0869565i −0.662291 0.749247i \(-0.730416\pi\)
0.749247 + 0.662291i \(0.230416\pi\)
\(24\) 0 0
\(25\) −25.0000 −1.00000
\(26\) 6.00000 0.230769
\(27\) 0 0
\(28\) −4.00000 + 4.00000i −0.142857 + 0.142857i
\(29\) 40.0000i 1.37931i −0.724138 0.689655i \(-0.757762\pi\)
0.724138 0.689655i \(-0.242238\pi\)
\(30\) 0 0
\(31\) 52.0000 1.67742 0.838710 0.544579i \(-0.183310\pi\)
0.838710 + 0.544579i \(0.183310\pi\)
\(32\) −4.00000 4.00000i −0.125000 0.125000i
\(33\) 0 0
\(34\) 14.0000i 0.411765i
\(35\) −10.0000 + 10.0000i −0.285714 + 0.285714i
\(36\) 0 0
\(37\) −3.00000 3.00000i −0.0810811 0.0810811i 0.665403 0.746484i \(-0.268260\pi\)
−0.746484 + 0.665403i \(0.768260\pi\)
\(38\) 20.0000 20.0000i 0.526316 0.526316i
\(39\) 0 0
\(40\) −10.0000 10.0000i −0.250000 0.250000i
\(41\) 8.00000 0.195122 0.0975610 0.995230i \(-0.468896\pi\)
0.0975610 + 0.995230i \(0.468896\pi\)
\(42\) 0 0
\(43\) −42.0000 + 42.0000i −0.976744 + 0.976744i −0.999736 0.0229915i \(-0.992681\pi\)
0.0229915 + 0.999736i \(0.492681\pi\)
\(44\) 16.0000i 0.363636i
\(45\) 0 0
\(46\) 4.00000 0.0869565
\(47\) 18.0000 + 18.0000i 0.382979 + 0.382979i 0.872174 0.489195i \(-0.162710\pi\)
−0.489195 + 0.872174i \(0.662710\pi\)
\(48\) 0 0
\(49\) 41.0000i 0.836735i
\(50\) −25.0000 25.0000i −0.500000 0.500000i
\(51\) 0 0
\(52\) 6.00000 + 6.00000i 0.115385 + 0.115385i
\(53\) −53.0000 + 53.0000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 40.0000i 0.727273i
\(56\) −8.00000 −0.142857
\(57\) 0 0
\(58\) 40.0000 40.0000i 0.689655 0.689655i
\(59\) 20.0000i 0.338983i 0.985532 + 0.169492i \(0.0542125\pi\)
−0.985532 + 0.169492i \(0.945787\pi\)
\(60\) 0 0
\(61\) −48.0000 −0.786885 −0.393443 0.919349i \(-0.628716\pi\)
−0.393443 + 0.919349i \(0.628716\pi\)
\(62\) 52.0000 + 52.0000i 0.838710 + 0.838710i
\(63\) 0 0
\(64\) 8.00000i 0.125000i
\(65\) 15.0000 + 15.0000i 0.230769 + 0.230769i
\(66\) 0 0
\(67\) 62.0000 + 62.0000i 0.925373 + 0.925373i 0.997403 0.0720294i \(-0.0229475\pi\)
−0.0720294 + 0.997403i \(0.522948\pi\)
\(68\) 14.0000 14.0000i 0.205882 0.205882i
\(69\) 0 0
\(70\) −20.0000 −0.285714
\(71\) 28.0000 0.394366 0.197183 0.980367i \(-0.436821\pi\)
0.197183 + 0.980367i \(0.436821\pi\)
\(72\) 0 0
\(73\) −47.0000 + 47.0000i −0.643836 + 0.643836i −0.951496 0.307661i \(-0.900454\pi\)
0.307661 + 0.951496i \(0.400454\pi\)
\(74\) 6.00000i 0.0810811i
\(75\) 0 0
\(76\) 40.0000 0.526316
\(77\) 16.0000 + 16.0000i 0.207792 + 0.207792i
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 20.0000i 0.250000i
\(81\) 0 0
\(82\) 8.00000 + 8.00000i 0.0975610 + 0.0975610i
\(83\) −18.0000 + 18.0000i −0.216867 + 0.216867i −0.807177 0.590310i \(-0.799006\pi\)
0.590310 + 0.807177i \(0.299006\pi\)
\(84\) 0 0
\(85\) 35.0000 35.0000i 0.411765 0.411765i
\(86\) −84.0000 −0.976744
\(87\) 0 0
\(88\) −16.0000 + 16.0000i −0.181818 + 0.181818i
\(89\) 80.0000i 0.898876i −0.893311 0.449438i \(-0.851624\pi\)
0.893311 0.449438i \(-0.148376\pi\)
\(90\) 0 0
\(91\) 12.0000 0.131868
\(92\) 4.00000 + 4.00000i 0.0434783 + 0.0434783i
\(93\) 0 0
\(94\) 36.0000i 0.382979i
\(95\) 100.000 1.05263
\(96\) 0 0
\(97\) −63.0000 63.0000i −0.649485 0.649485i 0.303384 0.952868i \(-0.401884\pi\)
−0.952868 + 0.303384i \(0.901884\pi\)
\(98\) 41.0000 41.0000i 0.418367 0.418367i
\(99\) 0 0
\(100\) 50.0000i 0.500000i
\(101\) −62.0000 −0.613861 −0.306931 0.951732i \(-0.599302\pi\)
−0.306931 + 0.951732i \(0.599302\pi\)
\(102\) 0 0
\(103\) 118.000 118.000i 1.14563 1.14563i 0.158229 0.987403i \(-0.449422\pi\)
0.987403 0.158229i \(-0.0505783\pi\)
\(104\) 12.0000i 0.115385i
\(105\) 0 0
\(106\) −106.000 −1.00000
\(107\) −142.000 142.000i −1.32710 1.32710i −0.907886 0.419217i \(-0.862305\pi\)
−0.419217 0.907886i \(-0.637695\pi\)
\(108\) 0 0
\(109\) 10.0000i 0.0917431i −0.998947 0.0458716i \(-0.985394\pi\)
0.998947 0.0458716i \(-0.0146065\pi\)
\(110\) −40.0000 + 40.0000i −0.363636 + 0.363636i
\(111\) 0 0
\(112\) −8.00000 8.00000i −0.0714286 0.0714286i
\(113\) −23.0000 + 23.0000i −0.203540 + 0.203540i −0.801515 0.597975i \(-0.795972\pi\)
0.597975 + 0.801515i \(0.295972\pi\)
\(114\) 0 0
\(115\) 10.0000 + 10.0000i 0.0869565 + 0.0869565i
\(116\) 80.0000 0.689655
\(117\) 0 0
\(118\) −20.0000 + 20.0000i −0.169492 + 0.169492i
\(119\) 28.0000i 0.235294i
\(120\) 0 0
\(121\) −57.0000 −0.471074
\(122\) −48.0000 48.0000i −0.393443 0.393443i
\(123\) 0 0
\(124\) 104.000i 0.838710i
\(125\) 125.000i 1.00000i
\(126\) 0 0
\(127\) −118.000 118.000i −0.929134 0.929134i 0.0685161 0.997650i \(-0.478174\pi\)
−0.997650 + 0.0685161i \(0.978174\pi\)
\(128\) 8.00000 8.00000i 0.0625000 0.0625000i
\(129\) 0 0
\(130\) 30.0000i 0.230769i
\(131\) 128.000 0.977099 0.488550 0.872536i \(-0.337526\pi\)
0.488550 + 0.872536i \(0.337526\pi\)
\(132\) 0 0
\(133\) 40.0000 40.0000i 0.300752 0.300752i
\(134\) 124.000i 0.925373i
\(135\) 0 0
\(136\) 28.0000 0.205882
\(137\) 63.0000 + 63.0000i 0.459854 + 0.459854i 0.898607 0.438753i \(-0.144580\pi\)
−0.438753 + 0.898607i \(0.644580\pi\)
\(138\) 0 0
\(139\) 140.000i 1.00719i 0.863939 + 0.503597i \(0.167990\pi\)
−0.863939 + 0.503597i \(0.832010\pi\)
\(140\) −20.0000 20.0000i −0.142857 0.142857i
\(141\) 0 0
\(142\) 28.0000 + 28.0000i 0.197183 + 0.197183i
\(143\) 24.0000 24.0000i 0.167832 0.167832i
\(144\) 0 0
\(145\) 200.000 1.37931
\(146\) −94.0000 −0.643836
\(147\) 0 0
\(148\) 6.00000 6.00000i 0.0405405 0.0405405i
\(149\) 150.000i 1.00671i 0.864079 + 0.503356i \(0.167901\pi\)
−0.864079 + 0.503356i \(0.832099\pi\)
\(150\) 0 0
\(151\) 52.0000 0.344371 0.172185 0.985065i \(-0.444917\pi\)
0.172185 + 0.985065i \(0.444917\pi\)
\(152\) 40.0000 + 40.0000i 0.263158 + 0.263158i
\(153\) 0 0
\(154\) 32.0000i 0.207792i
\(155\) 260.000i 1.67742i
\(156\) 0 0
\(157\) 27.0000 + 27.0000i 0.171975 + 0.171975i 0.787846 0.615872i \(-0.211196\pi\)
−0.615872 + 0.787846i \(0.711196\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 20.0000 20.0000i 0.125000 0.125000i
\(161\) 8.00000 0.0496894
\(162\) 0 0
\(163\) −82.0000 + 82.0000i −0.503067 + 0.503067i −0.912390 0.409322i \(-0.865765\pi\)
0.409322 + 0.912390i \(0.365765\pi\)
\(164\) 16.0000i 0.0975610i
\(165\) 0 0
\(166\) −36.0000 −0.216867
\(167\) −62.0000 62.0000i −0.371257 0.371257i 0.496678 0.867935i \(-0.334553\pi\)
−0.867935 + 0.496678i \(0.834553\pi\)
\(168\) 0 0
\(169\) 151.000i 0.893491i
\(170\) 70.0000 0.411765
\(171\) 0 0
\(172\) −84.0000 84.0000i −0.488372 0.488372i
\(173\) 107.000 107.000i 0.618497 0.618497i −0.326649 0.945146i \(-0.605919\pi\)
0.945146 + 0.326649i \(0.105919\pi\)
\(174\) 0 0
\(175\) −50.0000 50.0000i −0.285714 0.285714i
\(176\) −32.0000 −0.181818
\(177\) 0 0
\(178\) 80.0000 80.0000i 0.449438 0.449438i
\(179\) 220.000i 1.22905i 0.788897 + 0.614525i \(0.210652\pi\)
−0.788897 + 0.614525i \(0.789348\pi\)
\(180\) 0 0
\(181\) 2.00000 0.0110497 0.00552486 0.999985i \(-0.498241\pi\)
0.00552486 + 0.999985i \(0.498241\pi\)
\(182\) 12.0000 + 12.0000i 0.0659341 + 0.0659341i
\(183\) 0 0
\(184\) 8.00000i 0.0434783i
\(185\) 15.0000 15.0000i 0.0810811 0.0810811i
\(186\) 0 0
\(187\) −56.0000 56.0000i −0.299465 0.299465i
\(188\) −36.0000 + 36.0000i −0.191489 + 0.191489i
\(189\) 0 0
\(190\) 100.000 + 100.000i 0.526316 + 0.526316i
\(191\) −212.000 −1.10995 −0.554974 0.831868i \(-0.687272\pi\)
−0.554974 + 0.831868i \(0.687272\pi\)
\(192\) 0 0
\(193\) −57.0000 + 57.0000i −0.295337 + 0.295337i −0.839184 0.543847i \(-0.816967\pi\)
0.543847 + 0.839184i \(0.316967\pi\)
\(194\) 126.000i 0.649485i
\(195\) 0 0
\(196\) 82.0000 0.418367
\(197\) 3.00000 + 3.00000i 0.0152284 + 0.0152284i 0.714680 0.699452i \(-0.246572\pi\)
−0.699452 + 0.714680i \(0.746572\pi\)
\(198\) 0 0
\(199\) 120.000i 0.603015i −0.953464 0.301508i \(-0.902510\pi\)
0.953464 0.301508i \(-0.0974898\pi\)
\(200\) 50.0000 50.0000i 0.250000 0.250000i
\(201\) 0 0
\(202\) −62.0000 62.0000i −0.306931 0.306931i
\(203\) 80.0000 80.0000i 0.394089 0.394089i
\(204\) 0 0
\(205\) 40.0000i 0.195122i
\(206\) 236.000 1.14563
\(207\) 0 0
\(208\) −12.0000 + 12.0000i −0.0576923 + 0.0576923i
\(209\) 160.000i 0.765550i
\(210\) 0 0
\(211\) −328.000 −1.55450 −0.777251 0.629190i \(-0.783387\pi\)
−0.777251 + 0.629190i \(0.783387\pi\)
\(212\) −106.000 106.000i −0.500000 0.500000i
\(213\) 0 0
\(214\) 284.000i 1.32710i
\(215\) −210.000 210.000i −0.976744 0.976744i
\(216\) 0 0
\(217\) 104.000 + 104.000i 0.479263 + 0.479263i
\(218\) 10.0000 10.0000i 0.0458716 0.0458716i
\(219\) 0 0
\(220\) −80.0000 −0.363636
\(221\) −42.0000 −0.190045
\(222\) 0 0
\(223\) 138.000 138.000i 0.618834 0.618834i −0.326398 0.945232i \(-0.605835\pi\)
0.945232 + 0.326398i \(0.105835\pi\)
\(224\) 16.0000i 0.0714286i
\(225\) 0 0
\(226\) −46.0000 −0.203540
\(227\) −2.00000 2.00000i −0.00881057 0.00881057i 0.702688 0.711498i \(-0.251983\pi\)
−0.711498 + 0.702688i \(0.751983\pi\)
\(228\) 0 0
\(229\) 120.000i 0.524017i 0.965066 + 0.262009i \(0.0843849\pi\)
−0.965066 + 0.262009i \(0.915615\pi\)
\(230\) 20.0000i 0.0869565i
\(231\) 0 0
\(232\) 80.0000 + 80.0000i 0.344828 + 0.344828i
\(233\) −183.000 + 183.000i −0.785408 + 0.785408i −0.980738 0.195330i \(-0.937422\pi\)
0.195330 + 0.980738i \(0.437422\pi\)
\(234\) 0 0
\(235\) −90.0000 + 90.0000i −0.382979 + 0.382979i
\(236\) −40.0000 −0.169492
\(237\) 0 0
\(238\) 28.0000 28.0000i 0.117647 0.117647i
\(239\) 120.000i 0.502092i −0.967975 0.251046i \(-0.919225\pi\)
0.967975 0.251046i \(-0.0807746\pi\)
\(240\) 0 0
\(241\) 232.000 0.962656 0.481328 0.876541i \(-0.340155\pi\)
0.481328 + 0.876541i \(0.340155\pi\)
\(242\) −57.0000 57.0000i −0.235537 0.235537i
\(243\) 0 0
\(244\) 96.0000i 0.393443i
\(245\) 205.000 0.836735
\(246\) 0 0
\(247\) −60.0000 60.0000i −0.242915 0.242915i
\(248\) −104.000 + 104.000i −0.419355 + 0.419355i
\(249\) 0 0
\(250\) 125.000 125.000i 0.500000 0.500000i
\(251\) 48.0000 0.191235 0.0956175 0.995418i \(-0.469517\pi\)
0.0956175 + 0.995418i \(0.469517\pi\)
\(252\) 0 0
\(253\) 16.0000 16.0000i 0.0632411 0.0632411i
\(254\) 236.000i 0.929134i
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) 313.000 + 313.000i 1.21790 + 1.21790i 0.968366 + 0.249532i \(0.0802769\pi\)
0.249532 + 0.968366i \(0.419723\pi\)
\(258\) 0 0
\(259\) 12.0000i 0.0463320i
\(260\) −30.0000 + 30.0000i −0.115385 + 0.115385i
\(261\) 0 0
\(262\) 128.000 + 128.000i 0.488550 + 0.488550i
\(263\) 262.000 262.000i 0.996198 0.996198i −0.00379508 0.999993i \(-0.501208\pi\)
0.999993 + 0.00379508i \(0.00120801\pi\)
\(264\) 0 0
\(265\) −265.000 265.000i −1.00000 1.00000i
\(266\) 80.0000 0.300752
\(267\) 0 0
\(268\) −124.000 + 124.000i −0.462687 + 0.462687i
\(269\) 10.0000i 0.0371747i −0.999827 0.0185874i \(-0.994083\pi\)
0.999827 0.0185874i \(-0.00591688\pi\)
\(270\) 0 0
\(271\) 252.000 0.929889 0.464945 0.885340i \(-0.346074\pi\)
0.464945 + 0.885340i \(0.346074\pi\)
\(272\) 28.0000 + 28.0000i 0.102941 + 0.102941i
\(273\) 0 0
\(274\) 126.000i 0.459854i
\(275\) −200.000 −0.727273
\(276\) 0 0
\(277\) 267.000 + 267.000i 0.963899 + 0.963899i 0.999371 0.0354718i \(-0.0112934\pi\)
−0.0354718 + 0.999371i \(0.511293\pi\)
\(278\) −140.000 + 140.000i −0.503597 + 0.503597i
\(279\) 0 0
\(280\) 40.0000i 0.142857i
\(281\) −312.000 −1.11032 −0.555160 0.831743i \(-0.687343\pi\)
−0.555160 + 0.831743i \(0.687343\pi\)
\(282\) 0 0
\(283\) −262.000 + 262.000i −0.925795 + 0.925795i −0.997431 0.0716358i \(-0.977178\pi\)
0.0716358 + 0.997431i \(0.477178\pi\)
\(284\) 56.0000i 0.197183i
\(285\) 0 0
\(286\) 48.0000 0.167832
\(287\) 16.0000 + 16.0000i 0.0557491 + 0.0557491i
\(288\) 0 0
\(289\) 191.000i 0.660900i
\(290\) 200.000 + 200.000i 0.689655 + 0.689655i
\(291\) 0 0
\(292\) −94.0000 94.0000i −0.321918 0.321918i
\(293\) −243.000 + 243.000i −0.829352 + 0.829352i −0.987427 0.158075i \(-0.949471\pi\)
0.158075 + 0.987427i \(0.449471\pi\)
\(294\) 0 0
\(295\) −100.000 −0.338983
\(296\) 12.0000 0.0405405
\(297\) 0 0
\(298\) −150.000 + 150.000i −0.503356 + 0.503356i
\(299\) 12.0000i 0.0401338i
\(300\) 0 0
\(301\) −168.000 −0.558140
\(302\) 52.0000 + 52.0000i 0.172185 + 0.172185i
\(303\) 0 0
\(304\) 80.0000i 0.263158i
\(305\) 240.000i 0.786885i
\(306\) 0 0
\(307\) −18.0000 18.0000i −0.0586319 0.0586319i 0.677183 0.735815i \(-0.263201\pi\)
−0.735815 + 0.677183i \(0.763201\pi\)
\(308\) −32.0000 + 32.0000i −0.103896 + 0.103896i
\(309\) 0 0
\(310\) −260.000 + 260.000i −0.838710 + 0.838710i
\(311\) 388.000 1.24759 0.623794 0.781589i \(-0.285590\pi\)
0.623794 + 0.781589i \(0.285590\pi\)
\(312\) 0 0
\(313\) 183.000 183.000i 0.584665 0.584665i −0.351517 0.936182i \(-0.614334\pi\)
0.936182 + 0.351517i \(0.114334\pi\)
\(314\) 54.0000i 0.171975i
\(315\) 0 0
\(316\) 0 0
\(317\) 213.000 + 213.000i 0.671924 + 0.671924i 0.958159 0.286235i \(-0.0924038\pi\)
−0.286235 + 0.958159i \(0.592404\pi\)
\(318\) 0 0
\(319\) 320.000i 1.00313i
\(320\) 40.0000 0.125000
\(321\) 0 0
\(322\) 8.00000 + 8.00000i 0.0248447 + 0.0248447i
\(323\) −140.000 + 140.000i −0.433437 + 0.433437i
\(324\) 0 0
\(325\) −75.0000 + 75.0000i −0.230769 + 0.230769i
\(326\) −164.000 −0.503067
\(327\) 0 0
\(328\) −16.0000 + 16.0000i −0.0487805 + 0.0487805i
\(329\) 72.0000i 0.218845i
\(330\) 0 0
\(331\) 232.000 0.700906 0.350453 0.936580i \(-0.386028\pi\)
0.350453 + 0.936580i \(0.386028\pi\)
\(332\) −36.0000 36.0000i −0.108434 0.108434i
\(333\) 0 0
\(334\) 124.000i 0.371257i
\(335\) −310.000 + 310.000i −0.925373 + 0.925373i
\(336\) 0 0
\(337\) 417.000 + 417.000i 1.23739 + 1.23739i 0.961064 + 0.276324i \(0.0891164\pi\)
0.276324 + 0.961064i \(0.410884\pi\)
\(338\) −151.000 + 151.000i −0.446746 + 0.446746i
\(339\) 0 0
\(340\) 70.0000 + 70.0000i 0.205882 + 0.205882i
\(341\) 416.000 1.21994
\(342\) 0 0
\(343\) 180.000 180.000i 0.524781 0.524781i
\(344\) 168.000i 0.488372i
\(345\) 0 0
\(346\) 214.000 0.618497
\(347\) −202.000 202.000i −0.582133 0.582133i 0.353356 0.935489i \(-0.385040\pi\)
−0.935489 + 0.353356i \(0.885040\pi\)
\(348\) 0 0
\(349\) 440.000i 1.26074i −0.776293 0.630372i \(-0.782902\pi\)
0.776293 0.630372i \(-0.217098\pi\)
\(350\) 100.000i 0.285714i
\(351\) 0 0
\(352\) −32.0000 32.0000i −0.0909091 0.0909091i
\(353\) 447.000 447.000i 1.26629 1.26629i 0.318298 0.947991i \(-0.396889\pi\)
0.947991 0.318298i \(-0.103111\pi\)
\(354\) 0 0
\(355\) 140.000i 0.394366i
\(356\) 160.000 0.449438
\(357\) 0 0
\(358\) −220.000 + 220.000i −0.614525 + 0.614525i
\(359\) 400.000i 1.11421i 0.830443 + 0.557103i \(0.188087\pi\)
−0.830443 + 0.557103i \(0.811913\pi\)
\(360\) 0 0
\(361\) −39.0000 −0.108033
\(362\) 2.00000 + 2.00000i 0.00552486 + 0.00552486i
\(363\) 0 0
\(364\) 24.0000i 0.0659341i
\(365\) −235.000 235.000i −0.643836 0.643836i
\(366\) 0 0
\(367\) −118.000 118.000i −0.321526 0.321526i 0.527826 0.849352i \(-0.323007\pi\)
−0.849352 + 0.527826i \(0.823007\pi\)
\(368\) −8.00000 + 8.00000i −0.0217391 + 0.0217391i
\(369\) 0 0
\(370\) 30.0000 0.0810811
\(371\) −212.000 −0.571429
\(372\) 0 0
\(373\) −107.000 + 107.000i −0.286863 + 0.286863i −0.835839 0.548975i \(-0.815018\pi\)
0.548975 + 0.835839i \(0.315018\pi\)
\(374\) 112.000i 0.299465i
\(375\) 0 0
\(376\) −72.0000 −0.191489
\(377\) −120.000 120.000i −0.318302 0.318302i
\(378\) 0 0
\(379\) 340.000i 0.897098i 0.893758 + 0.448549i \(0.148059\pi\)
−0.893758 + 0.448549i \(0.851941\pi\)
\(380\) 200.000i 0.526316i
\(381\) 0 0
\(382\) −212.000 212.000i −0.554974 0.554974i
\(383\) 342.000 342.000i 0.892950 0.892950i −0.101849 0.994800i \(-0.532476\pi\)
0.994800 + 0.101849i \(0.0324760\pi\)
\(384\) 0 0
\(385\) −80.0000 + 80.0000i −0.207792 + 0.207792i
\(386\) −114.000 −0.295337
\(387\) 0 0
\(388\) 126.000 126.000i 0.324742 0.324742i
\(389\) 390.000i 1.00257i −0.865282 0.501285i \(-0.832861\pi\)
0.865282 0.501285i \(-0.167139\pi\)
\(390\) 0 0
\(391\) −28.0000 −0.0716113
\(392\) 82.0000 + 82.0000i 0.209184 + 0.209184i
\(393\) 0 0
\(394\) 6.00000i 0.0152284i
\(395\) 0 0
\(396\) 0 0
\(397\) −323.000 323.000i −0.813602 0.813602i 0.171570 0.985172i \(-0.445116\pi\)
−0.985172 + 0.171570i \(0.945116\pi\)
\(398\) 120.000 120.000i 0.301508 0.301508i
\(399\) 0 0
\(400\) 100.000 0.250000
\(401\) −642.000 −1.60100 −0.800499 0.599334i \(-0.795432\pi\)
−0.800499 + 0.599334i \(0.795432\pi\)
\(402\) 0 0
\(403\) 156.000 156.000i 0.387097 0.387097i
\(404\) 124.000i 0.306931i
\(405\) 0 0
\(406\) 160.000 0.394089
\(407\) −24.0000 24.0000i −0.0589681 0.0589681i
\(408\) 0 0
\(409\) 150.000i 0.366748i −0.983043 0.183374i \(-0.941298\pi\)
0.983043 0.183374i \(-0.0587020\pi\)
\(410\) −40.0000 + 40.0000i −0.0975610 + 0.0975610i
\(411\) 0 0
\(412\) 236.000 + 236.000i 0.572816 + 0.572816i
\(413\) −40.0000 + 40.0000i −0.0968523 + 0.0968523i
\(414\) 0 0
\(415\) −90.0000 90.0000i −0.216867 0.216867i
\(416\) −24.0000 −0.0576923
\(417\) 0 0
\(418\) 160.000 160.000i 0.382775 0.382775i
\(419\) 300.000i 0.715990i −0.933723 0.357995i \(-0.883460\pi\)
0.933723 0.357995i \(-0.116540\pi\)
\(420\) 0 0
\(421\) −208.000 −0.494062 −0.247031 0.969008i \(-0.579455\pi\)
−0.247031 + 0.969008i \(0.579455\pi\)
\(422\) −328.000 328.000i −0.777251 0.777251i
\(423\) 0 0
\(424\) 212.000i 0.500000i
\(425\) 175.000 + 175.000i 0.411765 + 0.411765i
\(426\) 0 0
\(427\) −96.0000 96.0000i −0.224824 0.224824i
\(428\) 284.000 284.000i 0.663551 0.663551i
\(429\) 0 0
\(430\) 420.000i 0.976744i
\(431\) 788.000 1.82831 0.914153 0.405369i \(-0.132857\pi\)
0.914153 + 0.405369i \(0.132857\pi\)
\(432\) 0 0
\(433\) −367.000 + 367.000i −0.847575 + 0.847575i −0.989830 0.142255i \(-0.954565\pi\)
0.142255 + 0.989830i \(0.454565\pi\)
\(434\) 208.000i 0.479263i
\(435\) 0 0
\(436\) 20.0000 0.0458716
\(437\) −40.0000 40.0000i −0.0915332 0.0915332i
\(438\) 0 0
\(439\) 560.000i 1.27563i 0.770191 + 0.637813i \(0.220161\pi\)
−0.770191 + 0.637813i \(0.779839\pi\)
\(440\) −80.0000 80.0000i −0.181818 0.181818i
\(441\) 0 0
\(442\) −42.0000 42.0000i −0.0950226 0.0950226i
\(443\) −378.000 + 378.000i −0.853273 + 0.853273i −0.990535 0.137262i \(-0.956170\pi\)
0.137262 + 0.990535i \(0.456170\pi\)
\(444\) 0 0
\(445\) 400.000 0.898876
\(446\) 276.000 0.618834
\(447\) 0 0
\(448\) 16.0000 16.0000i 0.0357143 0.0357143i
\(449\) 410.000i 0.913140i 0.889687 + 0.456570i \(0.150922\pi\)
−0.889687 + 0.456570i \(0.849078\pi\)
\(450\) 0 0
\(451\) 64.0000 0.141907
\(452\) −46.0000 46.0000i −0.101770 0.101770i
\(453\) 0 0
\(454\) 4.00000i 0.00881057i
\(455\) 60.0000i 0.131868i
\(456\) 0 0
\(457\) −393.000 393.000i −0.859956 0.859956i 0.131376 0.991333i \(-0.458060\pi\)
−0.991333 + 0.131376i \(0.958060\pi\)
\(458\) −120.000 + 120.000i −0.262009 + 0.262009i
\(459\) 0 0
\(460\) −20.0000 + 20.0000i −0.0434783 + 0.0434783i
\(461\) −622.000 −1.34924 −0.674620 0.738165i \(-0.735693\pi\)
−0.674620 + 0.738165i \(0.735693\pi\)
\(462\) 0 0
\(463\) 278.000 278.000i 0.600432 0.600432i −0.339995 0.940427i \(-0.610425\pi\)
0.940427 + 0.339995i \(0.110425\pi\)
\(464\) 160.000i 0.344828i
\(465\) 0 0
\(466\) −366.000 −0.785408
\(467\) 38.0000 + 38.0000i 0.0813704 + 0.0813704i 0.746621 0.665250i \(-0.231675\pi\)
−0.665250 + 0.746621i \(0.731675\pi\)
\(468\) 0 0
\(469\) 248.000i 0.528785i
\(470\) −180.000 −0.382979
\(471\) 0 0
\(472\) −40.0000 40.0000i −0.0847458 0.0847458i
\(473\) −336.000 + 336.000i −0.710359 + 0.710359i
\(474\) 0 0
\(475\) 500.000i 1.05263i
\(476\) 56.0000 0.117647
\(477\) 0 0
\(478\) 120.000 120.000i 0.251046 0.251046i
\(479\) 440.000i 0.918580i 0.888286 + 0.459290i \(0.151896\pi\)
−0.888286 + 0.459290i \(0.848104\pi\)
\(480\) 0 0
\(481\) −18.0000 −0.0374220
\(482\) 232.000 + 232.000i 0.481328 + 0.481328i
\(483\) 0 0
\(484\) 114.000i 0.235537i
\(485\) 315.000 315.000i 0.649485 0.649485i
\(486\) 0 0
\(487\) 522.000 + 522.000i 1.07187 + 1.07187i 0.997209 + 0.0746595i \(0.0237870\pi\)
0.0746595 + 0.997209i \(0.476213\pi\)
\(488\) 96.0000 96.0000i 0.196721 0.196721i
\(489\) 0 0
\(490\) 205.000 + 205.000i 0.418367 + 0.418367i
\(491\) 328.000 0.668024 0.334012 0.942569i \(-0.391597\pi\)
0.334012 + 0.942569i \(0.391597\pi\)
\(492\) 0 0
\(493\) −280.000 + 280.000i −0.567951 + 0.567951i
\(494\) 120.000i 0.242915i
\(495\) 0 0
\(496\) −208.000 −0.419355
\(497\) 56.0000 + 56.0000i 0.112676 + 0.112676i
\(498\) 0 0
\(499\) 380.000i 0.761523i 0.924673 + 0.380762i \(0.124338\pi\)
−0.924673 + 0.380762i \(0.875662\pi\)
\(500\) 250.000 0.500000
\(501\) 0 0
\(502\) 48.0000 + 48.0000i 0.0956175 + 0.0956175i
\(503\) 42.0000 42.0000i 0.0834990 0.0834990i −0.664124 0.747623i \(-0.731195\pi\)
0.747623 + 0.664124i \(0.231195\pi\)
\(504\) 0 0
\(505\) 310.000i 0.613861i
\(506\) 32.0000 0.0632411
\(507\) 0 0
\(508\) 236.000 236.000i 0.464567 0.464567i
\(509\) 440.000i 0.864440i 0.901768 + 0.432220i \(0.142270\pi\)
−0.901768 + 0.432220i \(0.857730\pi\)
\(510\) 0 0
\(511\) −188.000 −0.367906
\(512\) 16.0000 + 16.0000i 0.0312500 + 0.0312500i
\(513\) 0 0
\(514\) 626.000i 1.21790i
\(515\) 590.000 + 590.000i 1.14563 + 1.14563i
\(516\) 0 0
\(517\) 144.000 + 144.000i 0.278530 + 0.278530i
\(518\) 12.0000 12.0000i 0.0231660 0.0231660i
\(519\) 0 0
\(520\) −60.0000 −0.115385
\(521\) 258.000 0.495202 0.247601 0.968862i \(-0.420358\pi\)
0.247601 + 0.968862i \(0.420358\pi\)
\(522\) 0 0
\(523\) 258.000 258.000i 0.493308 0.493308i −0.416039 0.909347i \(-0.636582\pi\)
0.909347 + 0.416039i \(0.136582\pi\)
\(524\) 256.000i 0.488550i
\(525\) 0 0
\(526\) 524.000 0.996198
\(527\) −364.000 364.000i −0.690702 0.690702i
\(528\) 0 0
\(529\) 521.000i 0.984877i
\(530\) 530.000i 1.00000i
\(531\) 0 0
\(532\) 80.0000 + 80.0000i 0.150376 + 0.150376i
\(533\) 24.0000 24.0000i 0.0450281 0.0450281i
\(534\) 0 0
\(535\) 710.000 710.000i 1.32710 1.32710i
\(536\) −248.000 −0.462687
\(537\) 0 0
\(538\) 10.0000 10.0000i 0.0185874 0.0185874i
\(539\) 328.000i 0.608534i
\(540\) 0 0
\(541\) −338.000 −0.624769 −0.312384 0.949956i \(-0.601128\pi\)
−0.312384 + 0.949956i \(0.601128\pi\)
\(542\) 252.000 + 252.000i 0.464945 + 0.464945i
\(543\) 0 0
\(544\) 56.0000i 0.102941i
\(545\) 50.0000 0.0917431
\(546\) 0 0
\(547\) −558.000 558.000i −1.02011 1.02011i −0.999794 0.0203161i \(-0.993533\pi\)
−0.0203161 0.999794i \(-0.506467\pi\)
\(548\) −126.000 + 126.000i −0.229927 + 0.229927i
\(549\) 0 0
\(550\) −200.000 200.000i −0.363636 0.363636i
\(551\) −800.000 −1.45191
\(552\) 0 0
\(553\) 0 0
\(554\) 534.000i 0.963899i
\(555\) 0 0
\(556\) −280.000 −0.503597
\(557\) 3.00000 + 3.00000i 0.00538600 + 0.00538600i 0.709795 0.704409i \(-0.248788\pi\)
−0.704409 + 0.709795i \(0.748788\pi\)
\(558\) 0 0
\(559\) 252.000i 0.450805i
\(560\) 40.0000 40.0000i 0.0714286 0.0714286i
\(561\) 0 0
\(562\) −312.000 312.000i −0.555160 0.555160i
\(563\) 42.0000 42.0000i 0.0746004 0.0746004i −0.668822 0.743422i \(-0.733201\pi\)
0.743422 + 0.668822i \(0.233201\pi\)
\(564\) 0 0
\(565\) −115.000 115.000i −0.203540 0.203540i
\(566\) −524.000 −0.925795
\(567\) 0 0
\(568\) −56.0000 + 56.0000i −0.0985915 + 0.0985915i
\(569\) 950.000i 1.66960i −0.550557 0.834798i \(-0.685584\pi\)
0.550557 0.834798i \(-0.314416\pi\)
\(570\) 0 0
\(571\) 392.000 0.686515 0.343257 0.939241i \(-0.388470\pi\)
0.343257 + 0.939241i \(0.388470\pi\)
\(572\) 48.0000 + 48.0000i 0.0839161 + 0.0839161i
\(573\) 0 0
\(574\) 32.0000i 0.0557491i
\(575\) −50.0000 + 50.0000i −0.0869565 + 0.0869565i
\(576\) 0 0
\(577\) −473.000 473.000i −0.819757 0.819757i 0.166315 0.986073i \(-0.446813\pi\)
−0.986073 + 0.166315i \(0.946813\pi\)
\(578\) 191.000 191.000i 0.330450 0.330450i
\(579\) 0 0
\(580\) 400.000i 0.689655i
\(581\) −72.0000 −0.123924
\(582\) 0 0
\(583\) −424.000 + 424.000i −0.727273 + 0.727273i
\(584\) 188.000i 0.321918i
\(585\) 0 0
\(586\) −486.000 −0.829352
\(587\) 198.000 + 198.000i 0.337308 + 0.337308i 0.855353 0.518045i \(-0.173340\pi\)
−0.518045 + 0.855353i \(0.673340\pi\)
\(588\) 0 0
\(589\) 1040.00i 1.76570i
\(590\) −100.000 100.000i −0.169492 0.169492i
\(591\) 0 0
\(592\) 12.0000 + 12.0000i 0.0202703 + 0.0202703i
\(593\) 47.0000 47.0000i 0.0792580 0.0792580i −0.666366 0.745624i \(-0.732151\pi\)
0.745624 + 0.666366i \(0.232151\pi\)
\(594\) 0 0
\(595\) 140.000 0.235294
\(596\) −300.000 −0.503356
\(597\) 0 0
\(598\) 12.0000 12.0000i 0.0200669 0.0200669i
\(599\) 520.000i 0.868114i −0.900886 0.434057i \(-0.857082\pi\)
0.900886 0.434057i \(-0.142918\pi\)
\(600\) 0 0
\(601\) −328.000 −0.545757 −0.272879 0.962048i \(-0.587976\pi\)
−0.272879 + 0.962048i \(0.587976\pi\)
\(602\) −168.000 168.000i −0.279070 0.279070i
\(603\) 0 0
\(604\) 104.000i 0.172185i
\(605\) 285.000i 0.471074i
\(606\) 0 0
\(607\) 462.000 + 462.000i 0.761120 + 0.761120i 0.976525 0.215405i \(-0.0691070\pi\)
−0.215405 + 0.976525i \(0.569107\pi\)
\(608\) −80.0000 + 80.0000i −0.131579 + 0.131579i
\(609\) 0 0
\(610\) 240.000 240.000i 0.393443 0.393443i
\(611\) 108.000 0.176759
\(612\) 0 0
\(613\) 723.000 723.000i 1.17945 1.17945i 0.199560 0.979886i \(-0.436049\pi\)
0.979886 0.199560i \(-0.0639512\pi\)
\(614\) 36.0000i 0.0586319i
\(615\) 0 0
\(616\) −64.0000 −0.103896
\(617\) −327.000 327.000i −0.529984 0.529984i 0.390584 0.920567i \(-0.372273\pi\)
−0.920567 + 0.390584i \(0.872273\pi\)
\(618\) 0 0
\(619\) 660.000i 1.06624i −0.846041 0.533118i \(-0.821020\pi\)
0.846041 0.533118i \(-0.178980\pi\)
\(620\) −520.000 −0.838710
\(621\) 0 0
\(622\) 388.000 + 388.000i 0.623794 + 0.623794i
\(623\) 160.000 160.000i 0.256822 0.256822i
\(624\) 0 0
\(625\) 625.000 1.00000
\(626\) 366.000 0.584665
\(627\) 0 0
\(628\) −54.0000 + 54.0000i −0.0859873 + 0.0859873i
\(629\) 42.0000i 0.0667727i
\(630\) 0 0
\(631\) −548.000 −0.868463 −0.434231 0.900801i \(-0.642980\pi\)
−0.434231 + 0.900801i \(0.642980\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 426.000i 0.671924i
\(635\) 590.000 590.000i 0.929134 0.929134i
\(636\) 0 0
\(637\) −123.000 123.000i −0.193093 0.193093i
\(638\) 320.000 320.000i 0.501567 0.501567i
\(639\) 0 0
\(640\) 40.0000 + 40.0000i 0.0625000 + 0.0625000i
\(641\) 568.000 0.886115 0.443058 0.896493i \(-0.353894\pi\)
0.443058 + 0.896493i \(0.353894\pi\)
\(642\) 0 0
\(643\) −342.000 + 342.000i −0.531882 + 0.531882i −0.921132 0.389250i \(-0.872734\pi\)
0.389250 + 0.921132i \(0.372734\pi\)
\(644\) 16.0000i 0.0248447i
\(645\) 0 0
\(646\) −280.000 −0.433437
\(647\) 118.000 + 118.000i 0.182380 + 0.182380i 0.792392 0.610012i \(-0.208835\pi\)
−0.610012 + 0.792392i \(0.708835\pi\)
\(648\) 0 0
\(649\) 160.000i 0.246533i
\(650\) −150.000 −0.230769
\(651\) 0 0
\(652\) −164.000 164.000i −0.251534 0.251534i
\(653\) −453.000 + 453.000i −0.693721 + 0.693721i −0.963049 0.269327i \(-0.913199\pi\)
0.269327 + 0.963049i \(0.413199\pi\)
\(654\) 0 0
\(655\) 640.000i 0.977099i
\(656\) −32.0000 −0.0487805
\(657\) 0 0
\(658\) −72.0000 + 72.0000i −0.109422 + 0.109422i
\(659\) 140.000i 0.212443i −0.994342 0.106222i \(-0.966125\pi\)
0.994342 0.106222i \(-0.0338753\pi\)
\(660\) 0 0
\(661\) 512.000 0.774584 0.387292 0.921957i \(-0.373411\pi\)
0.387292 + 0.921957i \(0.373411\pi\)
\(662\) 232.000 + 232.000i 0.350453 + 0.350453i
\(663\) 0 0
\(664\) 72.0000i 0.108434i
\(665\) 200.000 + 200.000i 0.300752 + 0.300752i
\(666\) 0 0
\(667\) −80.0000 80.0000i −0.119940 0.119940i
\(668\) 124.000 124.000i 0.185629 0.185629i
\(669\) 0 0
\(670\) −620.000 −0.925373
\(671\) −384.000 −0.572280
\(672\) 0 0
\(673\) 193.000 193.000i 0.286776 0.286776i −0.549028 0.835804i \(-0.685002\pi\)
0.835804 + 0.549028i \(0.185002\pi\)
\(674\) 834.000i 1.23739i
\(675\) 0 0
\(676\) −302.000 −0.446746
\(677\) −157.000 157.000i −0.231905 0.231905i 0.581582 0.813488i \(-0.302434\pi\)
−0.813488 + 0.581582i \(0.802434\pi\)
\(678\) 0 0
\(679\) 252.000i 0.371134i
\(680\) 140.000i 0.205882i
\(681\) 0 0
\(682\) 416.000 + 416.000i 0.609971 + 0.609971i
\(683\) −438.000 + 438.000i −0.641288 + 0.641288i −0.950872 0.309584i \(-0.899810\pi\)
0.309584 + 0.950872i \(0.399810\pi\)
\(684\) 0 0
\(685\) −315.000 + 315.000i −0.459854 + 0.459854i
\(686\) 360.000 0.524781
\(687\) 0 0
\(688\) 168.000 168.000i 0.244186 0.244186i
\(689\) 318.000i 0.461538i
\(690\) 0 0
\(691\) 1032.00 1.49349 0.746744 0.665112i \(-0.231616\pi\)
0.746744 + 0.665112i \(0.231616\pi\)
\(692\) 214.000 + 214.000i 0.309249 + 0.309249i
\(693\) 0 0
\(694\) 404.000i 0.582133i
\(695\) −700.000 −1.00719
\(696\) 0 0
\(697\) −56.0000 56.0000i −0.0803443 0.0803443i
\(698\) 440.000 440.000i 0.630372 0.630372i
\(699\) 0 0
\(700\) 100.000 100.000i 0.142857 0.142857i
\(701\) 128.000 0.182596 0.0912981 0.995824i \(-0.470898\pi\)
0.0912981 + 0.995824i \(0.470898\pi\)
\(702\) 0 0
\(703\) −60.0000 + 60.0000i −0.0853485 + 0.0853485i
\(704\) 64.0000i 0.0909091i
\(705\) 0 0
\(706\) 894.000 1.26629
\(707\) −124.000 124.000i −0.175389 0.175389i
\(708\) 0 0
\(709\) 760.000i 1.07193i 0.844239 + 0.535966i \(0.180053\pi\)
−0.844239 + 0.535966i \(0.819947\pi\)
\(710\) −140.000 + 140.000i −0.197183 + 0.197183i
\(711\) 0 0
\(712\) 160.000 + 160.000i 0.224719 + 0.224719i
\(713\) 104.000 104.000i 0.145863 0.145863i
\(714\) 0 0
\(715\) 120.000 + 120.000i 0.167832 + 0.167832i
\(716\) −440.000 −0.614525
\(717\) 0 0
\(718\) −400.000 + 400.000i −0.557103 + 0.557103i
\(719\) 1160.00i 1.61335i 0.590994 + 0.806676i \(0.298736\pi\)
−0.590994 + 0.806676i \(0.701264\pi\)
\(720\) 0 0
\(721\) 472.000 0.654646
\(722\) −39.0000 39.0000i −0.0540166 0.0540166i
\(723\) 0 0
\(724\) 4.00000i 0.00552486i
\(725\) 1000.00i 1.37931i
\(726\) 0 0
\(727\) −558.000 558.000i −0.767538 0.767538i 0.210135 0.977672i \(-0.432610\pi\)
−0.977672 + 0.210135i \(0.932610\pi\)
\(728\) −24.0000 + 24.0000i −0.0329670 + 0.0329670i
\(729\) 0 0
\(730\) 470.000i 0.643836i
\(731\) 588.000 0.804378
\(732\) 0 0
\(733\) −827.000 + 827.000i −1.12824 + 1.12824i −0.137777 + 0.990463i \(0.543996\pi\)
−0.990463 + 0.137777i \(0.956004\pi\)
\(734\) 236.000i 0.321526i
\(735\) 0 0
\(736\) −16.0000 −0.0217391
\(737\) 496.000 + 496.000i 0.672999 + 0.672999i
\(738\) 0 0
\(739\) 700.000i 0.947226i −0.880733 0.473613i \(-0.842950\pi\)
0.880733 0.473613i \(-0.157050\pi\)
\(740\) 30.0000 + 30.0000i 0.0405405 + 0.0405405i
\(741\) 0 0
\(742\) −212.000 212.000i −0.285714 0.285714i
\(743\) 382.000 382.000i 0.514132 0.514132i −0.401658 0.915790i \(-0.631566\pi\)
0.915790 + 0.401658i \(0.131566\pi\)
\(744\) 0 0
\(745\) −750.000 −1.00671
\(746\) −214.000 −0.286863
\(747\) 0 0
\(748\) 112.000 112.000i 0.149733 0.149733i
\(749\) 568.000i 0.758344i
\(750\) 0 0
\(751\) −588.000 −0.782956 −0.391478 0.920187i \(-0.628036\pi\)
−0.391478 + 0.920187i \(0.628036\pi\)
\(752\) −72.0000 72.0000i −0.0957447 0.0957447i
\(753\) 0 0
\(754\) 240.000i 0.318302i
\(755\) 260.000i 0.344371i
\(756\) 0 0
\(757\) 987.000 + 987.000i 1.30383 + 1.30383i 0.925788 + 0.378043i \(0.123403\pi\)
0.378043 + 0.925788i \(0.376597\pi\)
\(758\) −340.000 + 340.000i −0.448549 + 0.448549i
\(759\) 0 0
\(760\) −200.000 + 200.000i −0.263158 + 0.263158i
\(761\) 158.000 0.207622 0.103811 0.994597i \(-0.466896\pi\)
0.103811 + 0.994597i \(0.466896\pi\)
\(762\) 0 0
\(763\) 20.0000 20.0000i 0.0262123 0.0262123i
\(764\) 424.000i 0.554974i
\(765\) 0 0
\(766\) 684.000 0.892950
\(767\) 60.0000 + 60.0000i 0.0782269 + 0.0782269i
\(768\) 0 0
\(769\) 80.0000i 0.104031i 0.998646 + 0.0520156i \(0.0165646\pi\)
−0.998646 + 0.0520156i \(0.983435\pi\)
\(770\) −160.000 −0.207792
\(771\) 0 0
\(772\) −114.000 114.000i −0.147668 0.147668i
\(773\) −243.000 + 243.000i −0.314360 + 0.314360i −0.846596 0.532236i \(-0.821352\pi\)
0.532236 + 0.846596i \(0.321352\pi\)
\(774\) 0 0
\(775\) −1300.00 −1.67742
\(776\) 252.000 0.324742
\(777\) 0 0
\(778\) 390.000 390.000i 0.501285 0.501285i
\(779\) 160.000i 0.205392i
\(780\) 0 0
\(781\) 224.000 0.286812
\(782\) −28.0000 28.0000i −0.0358056 0.0358056i
\(783\) 0 0
\(784\) 164.000i 0.209184i
\(785\) −135.000 + 135.000i −0.171975 + 0.171975i
\(786\) 0 0
\(787\) 262.000 + 262.000i 0.332910 + 0.332910i 0.853690 0.520781i \(-0.174359\pi\)
−0.520781 + 0.853690i \(0.674359\pi\)
\(788\) −6.00000 + 6.00000i −0.00761421 + 0.00761421i
\(789\) 0 0
\(790\) 0 0
\(791\) −92.0000 −0.116308
\(792\) 0 0
\(793\) −144.000 + 144.000i −0.181589 + 0.181589i
\(794\) 646.000i 0.813602i
\(795\) 0 0
\(796\) 240.000 0.301508
\(797\) −267.000 267.000i −0.335006 0.335006i 0.519478 0.854484i \(-0.326127\pi\)
−0.854484 + 0.519478i \(0.826127\pi\)
\(798\) 0 0
\(799\) 252.000i 0.315394i
\(800\) 100.000 + 100.000i 0.125000 + 0.125000i
\(801\) 0 0
\(802\) −642.000 642.000i −0.800499 0.800499i
\(803\) −376.000 + 376.000i −0.468244 + 0.468244i
\(804\) 0 0
\(805\) 40.0000i 0.0496894i
\(806\) 312.000 0.387097
\(807\) 0 0
\(808\) 124.000 124.000i 0.153465 0.153465i
\(809\) 560.000i 0.692213i −0.938195 0.346106i \(-0.887504\pi\)
0.938195 0.346106i \(-0.112496\pi\)
\(810\) 0 0
\(811\) −208.000 −0.256473 −0.128237 0.991744i \(-0.540932\pi\)
−0.128237 + 0.991744i \(0.540932\pi\)
\(812\) 160.000 + 160.000i 0.197044 + 0.197044i
\(813\) 0 0
\(814\) 48.0000i 0.0589681i
\(815\) −410.000 410.000i −0.503067 0.503067i
\(816\) 0 0
\(817\) 840.000 + 840.000i 1.02815 + 1.02815i
\(818\) 150.000 150.000i 0.183374 0.183374i
\(819\) 0 0
\(820\) −80.0000 −0.0975610
\(821\) 1568.00 1.90987 0.954933 0.296821i \(-0.0959266\pi\)
0.954933 + 0.296821i \(0.0959266\pi\)
\(822\) 0 0
\(823\) −562.000 + 562.000i −0.682868 + 0.682868i −0.960645 0.277778i \(-0.910402\pi\)
0.277778 + 0.960645i \(0.410402\pi\)
\(824\) 472.000i 0.572816i
\(825\) 0 0
\(826\) −80.0000 −0.0968523
\(827\) −762.000 762.000i −0.921403 0.921403i 0.0757260 0.997129i \(-0.475873\pi\)
−0.997129 + 0.0757260i \(0.975873\pi\)
\(828\) 0 0
\(829\) 170.000i 0.205066i 0.994730 + 0.102533i \(0.0326948\pi\)
−0.994730 + 0.102533i \(0.967305\pi\)
\(830\) 180.000i 0.216867i
\(831\) 0 0
\(832\) −24.0000 24.0000i −0.0288462 0.0288462i
\(833\) −287.000 + 287.000i −0.344538 + 0.344538i
\(834\) 0 0
\(835\) 310.000 310.000i 0.371257 0.371257i
\(836\) 320.000 0.382775
\(837\) 0 0
\(838\) 300.000 300.000i 0.357995 0.357995i
\(839\) 280.000i 0.333731i 0.985980 + 0.166865i \(0.0533645\pi\)
−0.985980 + 0.166865i \(0.946635\pi\)
\(840\) 0 0
\(841\) −759.000 −0.902497
\(842\) −208.000 208.000i −0.247031 0.247031i
\(843\) 0 0
\(844\) 656.000i 0.777251i
\(845\) −755.000 −0.893491
\(846\) 0 0
\(847\) −114.000 114.000i −0.134593 0.134593i
\(848\) 212.000 212.000i 0.250000 0.250000i
\(849\) 0 0
\(850\) 350.000i 0.411765i
\(851\) −12.0000 −0.0141011
\(852\) 0 0
\(853\) 1123.00 1123.00i 1.31653 1.31653i 0.400026 0.916504i \(-0.369001\pi\)
0.916504 0.400026i \(-0.130999\pi\)
\(854\) 192.000i 0.224824i
\(855\) 0 0
\(856\) 568.000 0.663551
\(857\) −417.000 417.000i −0.486581 0.486581i 0.420644 0.907226i \(-0.361804\pi\)
−0.907226 + 0.420644i \(0.861804\pi\)
\(858\) 0 0
\(859\) 1300.00i 1.51339i −0.653769 0.756694i \(-0.726813\pi\)
0.653769 0.756694i \(-0.273187\pi\)
\(860\) 420.000 420.000i 0.488372 0.488372i
\(861\) 0 0
\(862\) 788.000 + 788.000i 0.914153 + 0.914153i
\(863\) 242.000 242.000i 0.280417 0.280417i −0.552858 0.833275i \(-0.686463\pi\)
0.833275 + 0.552858i \(0.186463\pi\)
\(864\) 0 0
\(865\) 535.000 + 535.000i 0.618497 + 0.618497i
\(866\) −734.000 −0.847575
\(867\) 0 0
\(868\) −208.000 + 208.000i −0.239631 + 0.239631i
\(869\) 0 0
\(870\) 0 0
\(871\) 372.000 0.427095
\(872\) 20.0000 + 20.0000i 0.0229358 + 0.0229358i
\(873\) 0 0
\(874\) 80.0000i 0.0915332i
\(875\) 250.000 250.000i 0.285714 0.285714i
\(876\) 0 0
\(877\) −453.000 453.000i −0.516534 0.516534i 0.399987 0.916521i \(-0.369015\pi\)
−0.916521 + 0.399987i \(0.869015\pi\)
\(878\) −560.000 + 560.000i −0.637813 + 0.637813i
\(879\) 0 0
\(880\) 160.000i 0.181818i
\(881\) −712.000 −0.808173 −0.404086 0.914721i \(-0.632410\pi\)
−0.404086 + 0.914721i \(0.632410\pi\)
\(882\) 0 0
\(883\) 118.000 118.000i 0.133635 0.133635i −0.637125 0.770760i \(-0.719877\pi\)
0.770760 + 0.637125i \(0.219877\pi\)
\(884\) 84.0000i 0.0950226i
\(885\) 0 0
\(886\) −756.000 −0.853273
\(887\) 1158.00 + 1158.00i 1.30552 + 1.30552i 0.924611 + 0.380914i \(0.124390\pi\)
0.380914 + 0.924611i \(0.375610\pi\)
\(888\) 0 0
\(889\) 472.000i 0.530934i
\(890\) 400.000 + 400.000i 0.449438 + 0.449438i
\(891\) 0 0
\(892\) 276.000 + 276.000i 0.309417 + 0.309417i
\(893\) 360.000 360.000i 0.403135 0.403135i
\(894\) 0 0
\(895\) −1100.00 −1.22905
\(896\) 32.0000 0.0357143
\(897\) 0 0
\(898\) −410.000 + 410.000i −0.456570 + 0.456570i
\(899\) 2080.00i 2.31368i
\(900\) 0 0
\(901\) 742.000 0.823529
\(902\) 64.0000 + 64.0000i 0.0709534 + 0.0709534i
\(903\) 0 0
\(904\) 92.0000i 0.101770i
\(905\) 10.0000i 0.0110497i
\(906\) 0 0
\(907\) 142.000 + 142.000i 0.156560 + 0.156560i 0.781040 0.624480i \(-0.214689\pi\)
−0.624480 + 0.781040i \(0.714689\pi\)
\(908\) 4.00000 4.00000i 0.00440529 0.00440529i
\(909\) 0 0
\(910\) −60.0000 + 60.0000i −0.0659341 + 0.0659341i
\(911\) −1172.00 −1.28650 −0.643249 0.765657i \(-0.722414\pi\)
−0.643249 + 0.765657i \(0.722414\pi\)
\(912\) 0 0
\(913\) −144.000 + 144.000i −0.157722 + 0.157722i
\(914\) 786.000i 0.859956i
\(915\) 0 0
\(916\) −240.000 −0.262009
\(917\) 256.000 + 256.000i 0.279171 + 0.279171i
\(918\) 0 0
\(919\) 920.000i 1.00109i 0.865711 + 0.500544i \(0.166867\pi\)
−0.865711 + 0.500544i \(0.833133\pi\)
\(920\) −40.0000 −0.0434783
\(921\) 0 0
\(922\) −622.000 622.000i −0.674620 0.674620i
\(923\) 84.0000 84.0000i 0.0910076 0.0910076i
\(924\) 0 0
\(925\) 75.0000 + 75.0000i 0.0810811 + 0.0810811i
\(926\) 556.000 0.600432
\(927\) 0 0
\(928\) −160.000 + 160.000i −0.172414 + 0.172414i
\(929\) 1190.00i 1.28095i 0.767980 + 0.640474i \(0.221262\pi\)
−0.767980 + 0.640474i \(0.778738\pi\)
\(930\) 0 0
\(931\) −820.000 −0.880773
\(932\) −366.000 366.000i −0.392704 0.392704i
\(933\) 0 0
\(934\) 76.0000i 0.0813704i
\(935\) 280.000 280.000i 0.299465 0.299465i
\(936\) 0 0
\(937\) −233.000 233.000i −0.248666 0.248666i 0.571757 0.820423i \(-0.306262\pi\)
−0.820423 + 0.571757i \(0.806262\pi\)
\(938\) −248.000 + 248.000i −0.264392 + 0.264392i
\(939\) 0 0
\(940\) −180.000 180.000i −0.191489 0.191489i
\(941\) 78.0000 0.0828905 0.0414453 0.999141i \(-0.486804\pi\)
0.0414453 + 0.999141i \(0.486804\pi\)
\(942\) 0 0
\(943\) 16.0000 16.0000i 0.0169671 0.0169671i
\(944\) 80.0000i 0.0847458i
\(945\) 0 0
\(946\) −672.000 −0.710359
\(947\) −62.0000 62.0000i −0.0654699 0.0654699i 0.673614 0.739084i \(-0.264741\pi\)
−0.739084 + 0.673614i \(0.764741\pi\)
\(948\) 0 0
\(949\) 282.000i 0.297155i
\(950\) −500.000 + 500.000i −0.526316 + 0.526316i
\(951\) 0 0
\(952\) 56.0000 + 56.0000i 0.0588235 + 0.0588235i
\(953\) 1017.00 1017.00i 1.06716 1.06716i 0.0695800 0.997576i \(-0.477834\pi\)
0.997576 0.0695800i \(-0.0221659\pi\)
\(954\) 0 0
\(955\) 1060.00i 1.10995i
\(956\) 240.000 0.251046
\(957\) 0 0
\(958\) −440.000 + 440.000i −0.459290 + 0.459290i
\(959\) 252.000i 0.262774i
\(960\) 0 0
\(961\) 1743.00 1.81374
\(962\) −18.0000 18.0000i −0.0187110 0.0187110i
\(963\) 0 0
\(964\) 464.000i 0.481328i
\(965\) −285.000 285.000i −0.295337 0.295337i
\(966\) 0 0
\(967\) 502.000 + 502.000i 0.519131 + 0.519131i 0.917309 0.398177i \(-0.130357\pi\)
−0.398177 + 0.917309i \(0.630357\pi\)
\(968\) 114.000 114.000i 0.117769 0.117769i
\(969\) 0 0
\(970\) 630.000 0.649485
\(971\) −992.000 −1.02163 −0.510814 0.859691i \(-0.670656\pi\)
−0.510814 + 0.859691i \(0.670656\pi\)
\(972\) 0 0
\(973\) −280.000 + 280.000i −0.287770 + 0.287770i
\(974\) 1044.00i 1.07187i
\(975\) 0 0
\(976\) 192.000 0.196721
\(977\) 783.000 + 783.000i 0.801433 + 0.801433i 0.983320 0.181887i \(-0.0582204\pi\)
−0.181887 + 0.983320i \(0.558220\pi\)
\(978\) 0 0
\(979\) 640.000i 0.653728i
\(980\) 410.000i 0.418367i
\(981\) 0 0
\(982\) 328.000 + 328.000i 0.334012 + 0.334012i
\(983\) −1058.00 + 1058.00i −1.07630 + 1.07630i −0.0794589 + 0.996838i \(0.525319\pi\)
−0.996838 + 0.0794589i \(0.974681\pi\)
\(984\) 0 0
\(985\) −15.0000 + 15.0000i −0.0152284 + 0.0152284i
\(986\) −560.000 −0.567951
\(987\) 0 0
\(988\) 120.000 120.000i 0.121457 0.121457i
\(989\) 168.000i 0.169869i
\(990\) 0 0
\(991\) −68.0000 −0.0686176 −0.0343088 0.999411i \(-0.510923\pi\)
−0.0343088 + 0.999411i \(0.510923\pi\)
\(992\) −208.000 208.000i −0.209677 0.209677i
\(993\) 0 0
\(994\) 112.000i 0.112676i
\(995\) 600.000 0.603015
\(996\) 0 0
\(997\) −773.000 773.000i −0.775326 0.775326i 0.203706 0.979032i \(-0.434701\pi\)
−0.979032 + 0.203706i \(0.934701\pi\)
\(998\) −380.000 + 380.000i −0.380762 + 0.380762i
\(999\) 0 0
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 90.3.g.b.37.1 2
3.2 odd 2 10.3.c.a.7.1 yes 2
4.3 odd 2 720.3.bh.c.577.1 2
5.2 odd 4 450.3.g.b.343.1 2
5.3 odd 4 inner 90.3.g.b.73.1 2
5.4 even 2 450.3.g.b.307.1 2
12.11 even 2 80.3.p.c.17.1 2
15.2 even 4 50.3.c.c.43.1 2
15.8 even 4 10.3.c.a.3.1 2
15.14 odd 2 50.3.c.c.7.1 2
20.3 even 4 720.3.bh.c.433.1 2
21.20 even 2 490.3.f.b.197.1 2
24.5 odd 2 320.3.p.h.257.1 2
24.11 even 2 320.3.p.a.257.1 2
60.23 odd 4 80.3.p.c.33.1 2
60.47 odd 4 400.3.p.b.193.1 2
60.59 even 2 400.3.p.b.257.1 2
105.83 odd 4 490.3.f.b.393.1 2
120.53 even 4 320.3.p.h.193.1 2
120.83 odd 4 320.3.p.a.193.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.3.c.a.3.1 2 15.8 even 4
10.3.c.a.7.1 yes 2 3.2 odd 2
50.3.c.c.7.1 2 15.14 odd 2
50.3.c.c.43.1 2 15.2 even 4
80.3.p.c.17.1 2 12.11 even 2
80.3.p.c.33.1 2 60.23 odd 4
90.3.g.b.37.1 2 1.1 even 1 trivial
90.3.g.b.73.1 2 5.3 odd 4 inner
320.3.p.a.193.1 2 120.83 odd 4
320.3.p.a.257.1 2 24.11 even 2
320.3.p.h.193.1 2 120.53 even 4
320.3.p.h.257.1 2 24.5 odd 2
400.3.p.b.193.1 2 60.47 odd 4
400.3.p.b.257.1 2 60.59 even 2
450.3.g.b.307.1 2 5.4 even 2
450.3.g.b.343.1 2 5.2 odd 4
490.3.f.b.197.1 2 21.20 even 2
490.3.f.b.393.1 2 105.83 odd 4
720.3.bh.c.433.1 2 20.3 even 4
720.3.bh.c.577.1 2 4.3 odd 2