Properties

Label 450.4.f.a.143.1
Level $450$
Weight $4$
Character 450.143
Analytic conductor $26.551$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [450,4,Mod(107,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.107");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 450.f (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: \(26.5508595026\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 143.1
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 450.143
Dual form 450.4.f.a.107.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.41421 - 1.41421i) q^{2} +4.00000i q^{4} +(-9.00000 + 9.00000i) q^{7} +(5.65685 - 5.65685i) q^{8} +O(q^{10})\) \(q+(-1.41421 - 1.41421i) q^{2} +4.00000i q^{4} +(-9.00000 + 9.00000i) q^{7} +(5.65685 - 5.65685i) q^{8} -38.1838i q^{11} +(9.00000 + 9.00000i) q^{13} +25.4558 q^{14} -16.0000 q^{16} +(55.1543 + 55.1543i) q^{17} -38.0000i q^{19} +(-54.0000 + 54.0000i) q^{22} +(-29.6985 + 29.6985i) q^{23} -25.4558i q^{26} +(-36.0000 - 36.0000i) q^{28} +76.3675 q^{29} -236.000 q^{31} +(22.6274 + 22.6274i) q^{32} -156.000i q^{34} +(-279.000 + 279.000i) q^{37} +(-53.7401 + 53.7401i) q^{38} -343.654i q^{41} +(-360.000 - 360.000i) q^{43} +152.735 q^{44} +84.0000 q^{46} +(254.558 + 254.558i) q^{47} +181.000i q^{49} +(-36.0000 + 36.0000i) q^{52} +(-352.139 + 352.139i) q^{53} +101.823i q^{56} +(-108.000 - 108.000i) q^{58} -343.654 q^{59} +110.000 q^{61} +(333.754 + 333.754i) q^{62} -64.0000i q^{64} +(-702.000 + 702.000i) q^{67} +(-220.617 + 220.617i) q^{68} -76.3675i q^{71} +(-378.000 - 378.000i) q^{73} +789.131 q^{74} +152.000 q^{76} +(343.654 + 343.654i) q^{77} +272.000i q^{79} +(-486.000 + 486.000i) q^{82} +(59.3970 - 59.3970i) q^{83} +1018.23i q^{86} +(-216.000 - 216.000i) q^{88} -1489.17 q^{89} -162.000 q^{91} +(-118.794 - 118.794i) q^{92} -720.000i q^{94} +(-684.000 + 684.000i) q^{97} +(255.973 - 255.973i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 36 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 36 q^{7} + 36 q^{13} - 64 q^{16} - 216 q^{22} - 144 q^{28} - 944 q^{31} - 1116 q^{37} - 1440 q^{43} + 336 q^{46} - 144 q^{52} - 432 q^{58} + 440 q^{61} - 2808 q^{67} - 1512 q^{73} + 608 q^{76} - 1944 q^{82} - 864 q^{88} - 648 q^{91} - 2736 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 1.41421i −0.500000 0.500000i
\(3\) 0 0
\(4\) 4.00000i 0.500000i
\(5\) 0 0
\(6\) 0 0
\(7\) −9.00000 + 9.00000i −0.485954 + 0.485954i −0.907027 0.421073i \(-0.861654\pi\)
0.421073 + 0.907027i \(0.361654\pi\)
\(8\) 5.65685 5.65685i 0.250000 0.250000i
\(9\) 0 0
\(10\) 0 0
\(11\) 38.1838i 1.04662i −0.852142 0.523311i \(-0.824697\pi\)
0.852142 0.523311i \(-0.175303\pi\)
\(12\) 0 0
\(13\) 9.00000 + 9.00000i 0.192012 + 0.192012i 0.796565 0.604553i \(-0.206648\pi\)
−0.604553 + 0.796565i \(0.706648\pi\)
\(14\) 25.4558 0.485954
\(15\) 0 0
\(16\) −16.0000 −0.250000
\(17\) 55.1543 + 55.1543i 0.786876 + 0.786876i 0.980981 0.194105i \(-0.0621802\pi\)
−0.194105 + 0.980981i \(0.562180\pi\)
\(18\) 0 0
\(19\) 38.0000i 0.458831i −0.973329 0.229416i \(-0.926318\pi\)
0.973329 0.229416i \(-0.0736815\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −54.0000 + 54.0000i −0.523311 + 0.523311i
\(23\) −29.6985 + 29.6985i −0.269242 + 0.269242i −0.828795 0.559553i \(-0.810973\pi\)
0.559553 + 0.828795i \(0.310973\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 25.4558i 0.192012i
\(27\) 0 0
\(28\) −36.0000 36.0000i −0.242977 0.242977i
\(29\) 76.3675 0.489003 0.244502 0.969649i \(-0.421376\pi\)
0.244502 + 0.969649i \(0.421376\pi\)
\(30\) 0 0
\(31\) −236.000 −1.36732 −0.683659 0.729802i \(-0.739612\pi\)
−0.683659 + 0.729802i \(0.739612\pi\)
\(32\) 22.6274 + 22.6274i 0.125000 + 0.125000i
\(33\) 0 0
\(34\) 156.000i 0.786876i
\(35\) 0 0
\(36\) 0 0
\(37\) −279.000 + 279.000i −1.23966 + 1.23966i −0.279516 + 0.960141i \(0.590174\pi\)
−0.960141 + 0.279516i \(0.909826\pi\)
\(38\) −53.7401 + 53.7401i −0.229416 + 0.229416i
\(39\) 0 0
\(40\) 0 0
\(41\) 343.654i 1.30902i −0.756054 0.654509i \(-0.772875\pi\)
0.756054 0.654509i \(-0.227125\pi\)
\(42\) 0 0
\(43\) −360.000 360.000i −1.27673 1.27673i −0.942486 0.334247i \(-0.891518\pi\)
−0.334247 0.942486i \(-0.608482\pi\)
\(44\) 152.735 0.523311
\(45\) 0 0
\(46\) 84.0000 0.269242
\(47\) 254.558 + 254.558i 0.790025 + 0.790025i 0.981498 0.191473i \(-0.0613265\pi\)
−0.191473 + 0.981498i \(0.561326\pi\)
\(48\) 0 0
\(49\) 181.000i 0.527697i
\(50\) 0 0
\(51\) 0 0
\(52\) −36.0000 + 36.0000i −0.0960058 + 0.0960058i
\(53\) −352.139 + 352.139i −0.912642 + 0.912642i −0.996479 0.0838373i \(-0.973282\pi\)
0.0838373 + 0.996479i \(0.473282\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 101.823i 0.242977i
\(57\) 0 0
\(58\) −108.000 108.000i −0.244502 0.244502i
\(59\) −343.654 −0.758304 −0.379152 0.925334i \(-0.623784\pi\)
−0.379152 + 0.925334i \(0.623784\pi\)
\(60\) 0 0
\(61\) 110.000 0.230886 0.115443 0.993314i \(-0.463171\pi\)
0.115443 + 0.993314i \(0.463171\pi\)
\(62\) 333.754 + 333.754i 0.683659 + 0.683659i
\(63\) 0 0
\(64\) 64.0000i 0.125000i
\(65\) 0 0
\(66\) 0 0
\(67\) −702.000 + 702.000i −1.28004 + 1.28004i −0.339403 + 0.940641i \(0.610225\pi\)
−0.940641 + 0.339403i \(0.889775\pi\)
\(68\) −220.617 + 220.617i −0.393438 + 0.393438i
\(69\) 0 0
\(70\) 0 0
\(71\) 76.3675i 0.127650i −0.997961 0.0638251i \(-0.979670\pi\)
0.997961 0.0638251i \(-0.0203300\pi\)
\(72\) 0 0
\(73\) −378.000 378.000i −0.606049 0.606049i 0.335862 0.941911i \(-0.390972\pi\)
−0.941911 + 0.335862i \(0.890972\pi\)
\(74\) 789.131 1.23966
\(75\) 0 0
\(76\) 152.000 0.229416
\(77\) 343.654 + 343.654i 0.508610 + 0.508610i
\(78\) 0 0
\(79\) 272.000i 0.387372i 0.981064 + 0.193686i \(0.0620443\pi\)
−0.981064 + 0.193686i \(0.937956\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −486.000 + 486.000i −0.654509 + 0.654509i
\(83\) 59.3970 59.3970i 0.0785502 0.0785502i −0.666740 0.745290i \(-0.732311\pi\)
0.745290 + 0.666740i \(0.232311\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1018.23i 1.27673i
\(87\) 0 0
\(88\) −216.000 216.000i −0.261655 0.261655i
\(89\) −1489.17 −1.77361 −0.886805 0.462143i \(-0.847081\pi\)
−0.886805 + 0.462143i \(0.847081\pi\)
\(90\) 0 0
\(91\) −162.000 −0.186618
\(92\) −118.794 118.794i −0.134621 0.134621i
\(93\) 0 0
\(94\) 720.000i 0.790025i
\(95\) 0 0
\(96\) 0 0
\(97\) −684.000 + 684.000i −0.715976 + 0.715976i −0.967779 0.251803i \(-0.918977\pi\)
0.251803 + 0.967779i \(0.418977\pi\)
\(98\) 255.973 255.973i 0.263848 0.263848i
\(99\) 0 0
\(100\) 0 0
\(101\) 840.043i 0.827598i 0.910368 + 0.413799i \(0.135798\pi\)
−0.910368 + 0.413799i \(0.864202\pi\)
\(102\) 0 0
\(103\) 81.0000 + 81.0000i 0.0774871 + 0.0774871i 0.744788 0.667301i \(-0.232551\pi\)
−0.667301 + 0.744788i \(0.732551\pi\)
\(104\) 101.823 0.0960058
\(105\) 0 0
\(106\) 996.000 0.912642
\(107\) 1018.23 + 1018.23i 0.919966 + 0.919966i 0.997026 0.0770603i \(-0.0245534\pi\)
−0.0770603 + 0.997026i \(0.524553\pi\)
\(108\) 0 0
\(109\) 1658.00i 1.45695i 0.685072 + 0.728475i \(0.259771\pi\)
−0.685072 + 0.728475i \(0.740229\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 144.000 144.000i 0.121489 0.121489i
\(113\) −470.933 + 470.933i −0.392050 + 0.392050i −0.875418 0.483367i \(-0.839414\pi\)
0.483367 + 0.875418i \(0.339414\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 305.470i 0.244502i
\(117\) 0 0
\(118\) 486.000 + 486.000i 0.379152 + 0.379152i
\(119\) −992.778 −0.764771
\(120\) 0 0
\(121\) −127.000 −0.0954170
\(122\) −155.563 155.563i −0.115443 0.115443i
\(123\) 0 0
\(124\) 944.000i 0.683659i
\(125\) 0 0
\(126\) 0 0
\(127\) 747.000 747.000i 0.521933 0.521933i −0.396222 0.918155i \(-0.629679\pi\)
0.918155 + 0.396222i \(0.129679\pi\)
\(128\) −90.5097 + 90.5097i −0.0625000 + 0.0625000i
\(129\) 0 0
\(130\) 0 0
\(131\) 1718.27i 1.14600i −0.819556 0.573000i \(-0.805780\pi\)
0.819556 0.573000i \(-0.194220\pi\)
\(132\) 0 0
\(133\) 342.000 + 342.000i 0.222971 + 0.222971i
\(134\) 1985.56 1.28004
\(135\) 0 0
\(136\) 624.000 0.393438
\(137\) −963.079 963.079i −0.600594 0.600594i 0.339876 0.940470i \(-0.389615\pi\)
−0.940470 + 0.339876i \(0.889615\pi\)
\(138\) 0 0
\(139\) 1636.00i 0.998300i 0.866516 + 0.499150i \(0.166354\pi\)
−0.866516 + 0.499150i \(0.833646\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −108.000 + 108.000i −0.0638251 + 0.0638251i
\(143\) 343.654 343.654i 0.200964 0.200964i
\(144\) 0 0
\(145\) 0 0
\(146\) 1069.15i 0.606049i
\(147\) 0 0
\(148\) −1116.00 1116.00i −0.619829 0.619829i
\(149\) −1527.35 −0.839768 −0.419884 0.907578i \(-0.637929\pi\)
−0.419884 + 0.907578i \(0.637929\pi\)
\(150\) 0 0
\(151\) −128.000 −0.0689834 −0.0344917 0.999405i \(-0.510981\pi\)
−0.0344917 + 0.999405i \(0.510981\pi\)
\(152\) −214.960 214.960i −0.114708 0.114708i
\(153\) 0 0
\(154\) 972.000i 0.508610i
\(155\) 0 0
\(156\) 0 0
\(157\) 1683.00 1683.00i 0.855529 0.855529i −0.135279 0.990808i \(-0.543193\pi\)
0.990808 + 0.135279i \(0.0431930\pi\)
\(158\) 384.666 384.666i 0.193686 0.193686i
\(159\) 0 0
\(160\) 0 0
\(161\) 534.573i 0.261678i
\(162\) 0 0
\(163\) 198.000 + 198.000i 0.0951445 + 0.0951445i 0.753077 0.657932i \(-0.228569\pi\)
−0.657932 + 0.753077i \(0.728569\pi\)
\(164\) 1374.62 0.654509
\(165\) 0 0
\(166\) −168.000 −0.0785502
\(167\) −2982.58 2982.58i −1.38203 1.38203i −0.841013 0.541015i \(-0.818040\pi\)
−0.541015 0.841013i \(-0.681960\pi\)
\(168\) 0 0
\(169\) 2035.00i 0.926263i
\(170\) 0 0
\(171\) 0 0
\(172\) 1440.00 1440.00i 0.638366 0.638366i
\(173\) 1684.33 1684.33i 0.740215 0.740215i −0.232405 0.972619i \(-0.574659\pi\)
0.972619 + 0.232405i \(0.0746594\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 610.940i 0.261655i
\(177\) 0 0
\(178\) 2106.00 + 2106.00i 0.886805 + 0.886805i
\(179\) 343.654 0.143497 0.0717483 0.997423i \(-0.477142\pi\)
0.0717483 + 0.997423i \(0.477142\pi\)
\(180\) 0 0
\(181\) −2522.00 −1.03568 −0.517842 0.855476i \(-0.673264\pi\)
−0.517842 + 0.855476i \(0.673264\pi\)
\(182\) 229.103 + 229.103i 0.0933089 + 0.0933089i
\(183\) 0 0
\(184\) 336.000i 0.134621i
\(185\) 0 0
\(186\) 0 0
\(187\) 2106.00 2106.00i 0.823561 0.823561i
\(188\) −1018.23 + 1018.23i −0.395012 + 0.395012i
\(189\) 0 0
\(190\) 0 0
\(191\) 4505.68i 1.70691i −0.521166 0.853455i \(-0.674503\pi\)
0.521166 0.853455i \(-0.325497\pi\)
\(192\) 0 0
\(193\) 1458.00 + 1458.00i 0.543778 + 0.543778i 0.924634 0.380856i \(-0.124371\pi\)
−0.380856 + 0.924634i \(0.624371\pi\)
\(194\) 1934.64 0.715976
\(195\) 0 0
\(196\) −724.000 −0.263848
\(197\) 110.309 + 110.309i 0.0398942 + 0.0398942i 0.726772 0.686878i \(-0.241019\pi\)
−0.686878 + 0.726772i \(0.741019\pi\)
\(198\) 0 0
\(199\) 700.000i 0.249355i −0.992197 0.124678i \(-0.960210\pi\)
0.992197 0.124678i \(-0.0397897\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 1188.00 1188.00i 0.413799 0.413799i
\(203\) −687.308 + 687.308i −0.237633 + 0.237633i
\(204\) 0 0
\(205\) 0 0
\(206\) 229.103i 0.0774871i
\(207\) 0 0
\(208\) −144.000 144.000i −0.0480029 0.0480029i
\(209\) −1450.98 −0.480223
\(210\) 0 0
\(211\) −4084.00 −1.33248 −0.666242 0.745736i \(-0.732098\pi\)
−0.666242 + 0.745736i \(0.732098\pi\)
\(212\) −1408.56 1408.56i −0.456321 0.456321i
\(213\) 0 0
\(214\) 2880.00i 0.919966i
\(215\) 0 0
\(216\) 0 0
\(217\) 2124.00 2124.00i 0.664454 0.664454i
\(218\) 2344.77 2344.77i 0.728475 0.728475i
\(219\) 0 0
\(220\) 0 0
\(221\) 992.778i 0.302179i
\(222\) 0 0
\(223\) 2637.00 + 2637.00i 0.791868 + 0.791868i 0.981798 0.189930i \(-0.0608260\pi\)
−0.189930 + 0.981798i \(0.560826\pi\)
\(224\) −407.294 −0.121489
\(225\) 0 0
\(226\) 1332.00 0.392050
\(227\) −1671.60 1671.60i −0.488758 0.488758i 0.419156 0.907914i \(-0.362326\pi\)
−0.907914 + 0.419156i \(0.862326\pi\)
\(228\) 0 0
\(229\) 5434.00i 1.56807i 0.620714 + 0.784037i \(0.286843\pi\)
−0.620714 + 0.784037i \(0.713157\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 432.000 432.000i 0.122251 0.122251i
\(233\) 4034.75 4034.75i 1.13444 1.13444i 0.145014 0.989430i \(-0.453677\pi\)
0.989430 0.145014i \(-0.0463227\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1374.62i 0.379152i
\(237\) 0 0
\(238\) 1404.00 + 1404.00i 0.382386 + 0.382386i
\(239\) 3589.27 0.971426 0.485713 0.874118i \(-0.338560\pi\)
0.485713 + 0.874118i \(0.338560\pi\)
\(240\) 0 0
\(241\) 6212.00 1.66037 0.830187 0.557485i \(-0.188234\pi\)
0.830187 + 0.557485i \(0.188234\pi\)
\(242\) 179.605 + 179.605i 0.0477085 + 0.0477085i
\(243\) 0 0
\(244\) 440.000i 0.115443i
\(245\) 0 0
\(246\) 0 0
\(247\) 342.000 342.000i 0.0881010 0.0881010i
\(248\) −1335.02 + 1335.02i −0.341829 + 0.341829i
\(249\) 0 0
\(250\) 0 0
\(251\) 1641.90i 0.412892i −0.978458 0.206446i \(-0.933810\pi\)
0.978458 0.206446i \(-0.0661898\pi\)
\(252\) 0 0
\(253\) 1134.00 + 1134.00i 0.281794 + 0.281794i
\(254\) −2112.84 −0.521933
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) −1981.31 1981.31i −0.480899 0.480899i 0.424520 0.905419i \(-0.360443\pi\)
−0.905419 + 0.424520i \(0.860443\pi\)
\(258\) 0 0
\(259\) 5022.00i 1.20483i
\(260\) 0 0
\(261\) 0 0
\(262\) −2430.00 + 2430.00i −0.573000 + 0.573000i
\(263\) 420.021 420.021i 0.0984777 0.0984777i −0.656152 0.754629i \(-0.727817\pi\)
0.754629 + 0.656152i \(0.227817\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 967.322i 0.222971i
\(267\) 0 0
\(268\) −2808.00 2808.00i −0.640022 0.640022i
\(269\) −76.3675 −0.0173093 −0.00865467 0.999963i \(-0.502755\pi\)
−0.00865467 + 0.999963i \(0.502755\pi\)
\(270\) 0 0
\(271\) −3580.00 −0.802471 −0.401235 0.915975i \(-0.631419\pi\)
−0.401235 + 0.915975i \(0.631419\pi\)
\(272\) −882.469 882.469i −0.196719 0.196719i
\(273\) 0 0
\(274\) 2724.00i 0.600594i
\(275\) 0 0
\(276\) 0 0
\(277\) 2277.00 2277.00i 0.493905 0.493905i −0.415629 0.909534i \(-0.636439\pi\)
0.909534 + 0.415629i \(0.136439\pi\)
\(278\) 2313.65 2313.65i 0.499150 0.499150i
\(279\) 0 0
\(280\) 0 0
\(281\) 5613.01i 1.19162i 0.803127 + 0.595808i \(0.203168\pi\)
−0.803127 + 0.595808i \(0.796832\pi\)
\(282\) 0 0
\(283\) 1206.00 + 1206.00i 0.253319 + 0.253319i 0.822330 0.569011i \(-0.192674\pi\)
−0.569011 + 0.822330i \(0.692674\pi\)
\(284\) 305.470 0.0638251
\(285\) 0 0
\(286\) −972.000 −0.200964
\(287\) 3092.89 + 3092.89i 0.636123 + 0.636123i
\(288\) 0 0
\(289\) 1171.00i 0.238347i
\(290\) 0 0
\(291\) 0 0
\(292\) 1512.00 1512.00i 0.303024 0.303024i
\(293\) −6932.47 + 6932.47i −1.38225 + 1.38225i −0.541641 + 0.840610i \(0.682197\pi\)
−0.840610 + 0.541641i \(0.817803\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 3156.52i 0.619829i
\(297\) 0 0
\(298\) 2160.00 + 2160.00i 0.419884 + 0.419884i
\(299\) −534.573 −0.103395
\(300\) 0 0
\(301\) 6480.00 1.24087
\(302\) 181.019 + 181.019i 0.0344917 + 0.0344917i
\(303\) 0 0
\(304\) 608.000i 0.114708i
\(305\) 0 0
\(306\) 0 0
\(307\) −2160.00 + 2160.00i −0.401556 + 0.401556i −0.878781 0.477225i \(-0.841643\pi\)
0.477225 + 0.878781i \(0.341643\pi\)
\(308\) −1374.62 + 1374.62i −0.254305 + 0.254305i
\(309\) 0 0
\(310\) 0 0
\(311\) 4658.42i 0.849372i 0.905341 + 0.424686i \(0.139616\pi\)
−0.905341 + 0.424686i \(0.860384\pi\)
\(312\) 0 0
\(313\) −5742.00 5742.00i −1.03692 1.03692i −0.999292 0.0376317i \(-0.988019\pi\)
−0.0376317 0.999292i \(-0.511981\pi\)
\(314\) −4760.24 −0.855529
\(315\) 0 0
\(316\) −1088.00 −0.193686
\(317\) 4183.24 + 4183.24i 0.741181 + 0.741181i 0.972805 0.231624i \(-0.0744040\pi\)
−0.231624 + 0.972805i \(0.574404\pi\)
\(318\) 0 0
\(319\) 2916.00i 0.511801i
\(320\) 0 0
\(321\) 0 0
\(322\) −756.000 + 756.000i −0.130839 + 0.130839i
\(323\) 2095.86 2095.86i 0.361043 0.361043i
\(324\) 0 0
\(325\) 0 0
\(326\) 560.029i 0.0951445i
\(327\) 0 0
\(328\) −1944.00 1944.00i −0.327254 0.327254i
\(329\) −4582.05 −0.767832
\(330\) 0 0
\(331\) 2126.00 0.353038 0.176519 0.984297i \(-0.443516\pi\)
0.176519 + 0.984297i \(0.443516\pi\)
\(332\) 237.588 + 237.588i 0.0392751 + 0.0392751i
\(333\) 0 0
\(334\) 8436.00i 1.38203i
\(335\) 0 0
\(336\) 0 0
\(337\) −5958.00 + 5958.00i −0.963065 + 0.963065i −0.999342 0.0362767i \(-0.988450\pi\)
0.0362767 + 0.999342i \(0.488450\pi\)
\(338\) −2877.92 + 2877.92i −0.463132 + 0.463132i
\(339\) 0 0
\(340\) 0 0
\(341\) 9011.37i 1.43106i
\(342\) 0 0
\(343\) −4716.00 4716.00i −0.742391 0.742391i
\(344\) −4072.94 −0.638366
\(345\) 0 0
\(346\) −4764.00 −0.740215
\(347\) −2112.84 2112.84i −0.326867 0.326867i 0.524527 0.851394i \(-0.324242\pi\)
−0.851394 + 0.524527i \(0.824242\pi\)
\(348\) 0 0
\(349\) 2594.00i 0.397861i −0.980014 0.198931i \(-0.936253\pi\)
0.980014 0.198931i \(-0.0637469\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 864.000 864.000i 0.130828 0.130828i
\(353\) 980.050 980.050i 0.147770 0.147770i −0.629351 0.777121i \(-0.716679\pi\)
0.777121 + 0.629351i \(0.216679\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 5956.67i 0.886805i
\(357\) 0 0
\(358\) −486.000 486.000i −0.0717483 0.0717483i
\(359\) 9011.37 1.32480 0.662398 0.749152i \(-0.269539\pi\)
0.662398 + 0.749152i \(0.269539\pi\)
\(360\) 0 0
\(361\) 5415.00 0.789474
\(362\) 3566.65 + 3566.65i 0.517842 + 0.517842i
\(363\) 0 0
\(364\) 648.000i 0.0933089i
\(365\) 0 0
\(366\) 0 0
\(367\) −5751.00 + 5751.00i −0.817983 + 0.817983i −0.985816 0.167833i \(-0.946323\pi\)
0.167833 + 0.985816i \(0.446323\pi\)
\(368\) 475.176 475.176i 0.0673105 0.0673105i
\(369\) 0 0
\(370\) 0 0
\(371\) 6338.51i 0.887005i
\(372\) 0 0
\(373\) −2205.00 2205.00i −0.306087 0.306087i 0.537302 0.843390i \(-0.319443\pi\)
−0.843390 + 0.537302i \(0.819443\pi\)
\(374\) −5956.67 −0.823561
\(375\) 0 0
\(376\) 2880.00 0.395012
\(377\) 687.308 + 687.308i 0.0938943 + 0.0938943i
\(378\) 0 0
\(379\) 740.000i 0.100294i −0.998742 0.0501468i \(-0.984031\pi\)
0.998742 0.0501468i \(-0.0159689\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −6372.00 + 6372.00i −0.853455 + 0.853455i
\(383\) −8281.63 + 8281.63i −1.10489 + 1.10489i −0.111075 + 0.993812i \(0.535429\pi\)
−0.993812 + 0.111075i \(0.964571\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4123.85i 0.543778i
\(387\) 0 0
\(388\) −2736.00 2736.00i −0.357988 0.357988i
\(389\) 6109.40 0.796296 0.398148 0.917321i \(-0.369653\pi\)
0.398148 + 0.917321i \(0.369653\pi\)
\(390\) 0 0
\(391\) −3276.00 −0.423720
\(392\) 1023.89 + 1023.89i 0.131924 + 0.131924i
\(393\) 0 0
\(394\) 312.000i 0.0398942i
\(395\) 0 0
\(396\) 0 0
\(397\) −6849.00 + 6849.00i −0.865847 + 0.865847i −0.992010 0.126162i \(-0.959734\pi\)
0.126162 + 0.992010i \(0.459734\pi\)
\(398\) −989.949 + 989.949i −0.124678 + 0.124678i
\(399\) 0 0
\(400\) 0 0
\(401\) 11722.4i 1.45982i 0.683541 + 0.729912i \(0.260439\pi\)
−0.683541 + 0.729912i \(0.739561\pi\)
\(402\) 0 0
\(403\) −2124.00 2124.00i −0.262541 0.262541i
\(404\) −3360.17 −0.413799
\(405\) 0 0
\(406\) 1944.00 0.237633
\(407\) 10653.3 + 10653.3i 1.29745 + 1.29745i
\(408\) 0 0
\(409\) 4354.00i 0.526385i 0.964743 + 0.263192i \(0.0847754\pi\)
−0.964743 + 0.263192i \(0.915225\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −324.000 + 324.000i −0.0387435 + 0.0387435i
\(413\) 3092.89 3092.89i 0.368501 0.368501i
\(414\) 0 0
\(415\) 0 0
\(416\) 407.294i 0.0480029i
\(417\) 0 0
\(418\) 2052.00 + 2052.00i 0.240111 + 0.240111i
\(419\) 13708.0 1.59828 0.799139 0.601147i \(-0.205289\pi\)
0.799139 + 0.601147i \(0.205289\pi\)
\(420\) 0 0
\(421\) 8530.00 0.987474 0.493737 0.869611i \(-0.335630\pi\)
0.493737 + 0.869611i \(0.335630\pi\)
\(422\) 5775.65 + 5775.65i 0.666242 + 0.666242i
\(423\) 0 0
\(424\) 3984.00i 0.456321i
\(425\) 0 0
\(426\) 0 0
\(427\) −990.000 + 990.000i −0.112200 + 0.112200i
\(428\) −4072.94 + 4072.94i −0.459983 + 0.459983i
\(429\) 0 0
\(430\) 0 0
\(431\) 13593.4i 1.51919i −0.650395 0.759596i \(-0.725397\pi\)
0.650395 0.759596i \(-0.274603\pi\)
\(432\) 0 0
\(433\) 4626.00 + 4626.00i 0.513421 + 0.513421i 0.915573 0.402152i \(-0.131738\pi\)
−0.402152 + 0.915573i \(0.631738\pi\)
\(434\) −6007.58 −0.664454
\(435\) 0 0
\(436\) −6632.00 −0.728475
\(437\) 1128.54 + 1128.54i 0.123537 + 0.123537i
\(438\) 0 0
\(439\) 14092.0i 1.53206i −0.642805 0.766030i \(-0.722229\pi\)
0.642805 0.766030i \(-0.277771\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1404.00 1404.00i 0.151089 0.151089i
\(443\) 2664.38 2664.38i 0.285753 0.285753i −0.549645 0.835398i \(-0.685237\pi\)
0.835398 + 0.549645i \(0.185237\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 7458.56i 0.791868i
\(447\) 0 0
\(448\) 576.000 + 576.000i 0.0607443 + 0.0607443i
\(449\) −1260.06 −0.132441 −0.0662206 0.997805i \(-0.521094\pi\)
−0.0662206 + 0.997805i \(0.521094\pi\)
\(450\) 0 0
\(451\) −13122.0 −1.37005
\(452\) −1883.73 1883.73i −0.196025 0.196025i
\(453\) 0 0
\(454\) 4728.00i 0.488758i
\(455\) 0 0
\(456\) 0 0
\(457\) −8280.00 + 8280.00i −0.847532 + 0.847532i −0.989825 0.142293i \(-0.954553\pi\)
0.142293 + 0.989825i \(0.454553\pi\)
\(458\) 7684.84 7684.84i 0.784037 0.784037i
\(459\) 0 0
\(460\) 0 0
\(461\) 12371.5i 1.24989i 0.780668 + 0.624946i \(0.214879\pi\)
−0.780668 + 0.624946i \(0.785121\pi\)
\(462\) 0 0
\(463\) −8847.00 8847.00i −0.888024 0.888024i 0.106309 0.994333i \(-0.466097\pi\)
−0.994333 + 0.106309i \(0.966097\pi\)
\(464\) −1221.88 −0.122251
\(465\) 0 0
\(466\) −11412.0 −1.13444
\(467\) 4946.92 + 4946.92i 0.490184 + 0.490184i 0.908364 0.418180i \(-0.137332\pi\)
−0.418180 + 0.908364i \(0.637332\pi\)
\(468\) 0 0
\(469\) 12636.0i 1.24409i
\(470\) 0 0
\(471\) 0 0
\(472\) −1944.00 + 1944.00i −0.189576 + 0.189576i
\(473\) −13746.2 + 13746.2i −1.33626 + 1.33626i
\(474\) 0 0
\(475\) 0 0
\(476\) 3971.11i 0.382386i
\(477\) 0 0
\(478\) −5076.00 5076.00i −0.485713 0.485713i
\(479\) 12906.1 1.23110 0.615548 0.788099i \(-0.288935\pi\)
0.615548 + 0.788099i \(0.288935\pi\)
\(480\) 0 0
\(481\) −5022.00 −0.476057
\(482\) −8785.09 8785.09i −0.830187 0.830187i
\(483\) 0 0
\(484\) 508.000i 0.0477085i
\(485\) 0 0
\(486\) 0 0
\(487\) −369.000 + 369.000i −0.0343347 + 0.0343347i −0.724066 0.689731i \(-0.757729\pi\)
0.689731 + 0.724066i \(0.257729\pi\)
\(488\) 622.254 622.254i 0.0577215 0.0577215i
\(489\) 0 0
\(490\) 0 0
\(491\) 19817.4i 1.82148i −0.412983 0.910739i \(-0.635513\pi\)
0.412983 0.910739i \(-0.364487\pi\)
\(492\) 0 0
\(493\) 4212.00 + 4212.00i 0.384785 + 0.384785i
\(494\) −967.322 −0.0881010
\(495\) 0 0
\(496\) 3776.00 0.341829
\(497\) 687.308 + 687.308i 0.0620321 + 0.0620321i
\(498\) 0 0
\(499\) 3058.00i 0.274338i −0.990548 0.137169i \(-0.956200\pi\)
0.990548 0.137169i \(-0.0438004\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −2322.00 + 2322.00i −0.206446 + 0.206446i
\(503\) 11336.3 11336.3i 1.00490 1.00490i 0.00490714 0.999988i \(-0.498438\pi\)
0.999988 0.00490714i \(-0.00156200\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 3207.44i 0.281794i
\(507\) 0 0
\(508\) 2988.00 + 2988.00i 0.260967 + 0.260967i
\(509\) −13517.1 −1.17708 −0.588539 0.808469i \(-0.700297\pi\)
−0.588539 + 0.808469i \(0.700297\pi\)
\(510\) 0 0
\(511\) 6804.00 0.589024
\(512\) −362.039 362.039i −0.0312500 0.0312500i
\(513\) 0 0
\(514\) 5604.00i 0.480899i
\(515\) 0 0
\(516\) 0 0
\(517\) 9720.00 9720.00i 0.826857 0.826857i
\(518\) −7102.18 + 7102.18i −0.602417 + 0.602417i
\(519\) 0 0
\(520\) 0 0
\(521\) 8667.71i 0.728867i 0.931229 + 0.364433i \(0.118737\pi\)
−0.931229 + 0.364433i \(0.881263\pi\)
\(522\) 0 0
\(523\) 1242.00 + 1242.00i 0.103841 + 0.103841i 0.757119 0.653278i \(-0.226606\pi\)
−0.653278 + 0.757119i \(0.726606\pi\)
\(524\) 6873.08 0.573000
\(525\) 0 0
\(526\) −1188.00 −0.0984777
\(527\) −13016.4 13016.4i −1.07591 1.07591i
\(528\) 0 0
\(529\) 10403.0i 0.855018i
\(530\) 0 0
\(531\) 0 0
\(532\) −1368.00 + 1368.00i −0.111486 + 0.111486i
\(533\) 3092.89 3092.89i 0.251347 0.251347i
\(534\) 0 0
\(535\) 0 0
\(536\) 7942.22i 0.640022i
\(537\) 0 0
\(538\) 108.000 + 108.000i 0.00865467 + 0.00865467i
\(539\) 6911.26 0.552299
\(540\) 0 0
\(541\) 3242.00 0.257642 0.128821 0.991668i \(-0.458881\pi\)
0.128821 + 0.991668i \(0.458881\pi\)
\(542\) 5062.88 + 5062.88i 0.401235 + 0.401235i
\(543\) 0 0
\(544\) 2496.00i 0.196719i
\(545\) 0 0
\(546\) 0 0
\(547\) 10476.0 10476.0i 0.818869 0.818869i −0.167075 0.985944i \(-0.553432\pi\)
0.985944 + 0.167075i \(0.0534322\pi\)
\(548\) 3852.32 3852.32i 0.300297 0.300297i
\(549\) 0 0
\(550\) 0 0
\(551\) 2901.97i 0.224370i
\(552\) 0 0
\(553\) −2448.00 2448.00i −0.188245 0.188245i
\(554\) −6440.33 −0.493905
\(555\) 0 0
\(556\) −6544.00 −0.499150
\(557\) 165.463 + 165.463i 0.0125869 + 0.0125869i 0.713372 0.700785i \(-0.247167\pi\)
−0.700785 + 0.713372i \(0.747167\pi\)
\(558\) 0 0
\(559\) 6480.00i 0.490295i
\(560\) 0 0
\(561\) 0 0
\(562\) 7938.00 7938.00i 0.595808 0.595808i
\(563\) −14586.2 + 14586.2i −1.09189 + 1.09189i −0.0965649 + 0.995327i \(0.530786\pi\)
−0.995327 + 0.0965649i \(0.969214\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 3411.08i 0.253319i
\(567\) 0 0
\(568\) −432.000 432.000i −0.0319125 0.0319125i
\(569\) 7216.73 0.531707 0.265853 0.964013i \(-0.414346\pi\)
0.265853 + 0.964013i \(0.414346\pi\)
\(570\) 0 0
\(571\) −1762.00 −0.129137 −0.0645687 0.997913i \(-0.520567\pi\)
−0.0645687 + 0.997913i \(0.520567\pi\)
\(572\) 1374.62 + 1374.62i 0.100482 + 0.100482i
\(573\) 0 0
\(574\) 8748.00i 0.636123i
\(575\) 0 0
\(576\) 0 0
\(577\) 14688.0 14688.0i 1.05974 1.05974i 0.0616407 0.998098i \(-0.480367\pi\)
0.998098 0.0616407i \(-0.0196333\pi\)
\(578\) 1656.04 1656.04i 0.119174 0.119174i
\(579\) 0 0
\(580\) 0 0
\(581\) 1069.15i 0.0763436i
\(582\) 0 0
\(583\) 13446.0 + 13446.0i 0.955191 + 0.955191i
\(584\) −4276.58 −0.303024
\(585\) 0 0
\(586\) 19608.0 1.38225
\(587\) −2324.97 2324.97i −0.163478 0.163478i 0.620627 0.784106i \(-0.286878\pi\)
−0.784106 + 0.620627i \(0.786878\pi\)
\(588\) 0 0
\(589\) 8968.00i 0.627368i
\(590\) 0 0
\(591\) 0 0
\(592\) 4464.00 4464.00i 0.309914 0.309914i
\(593\) 15625.6 15625.6i 1.08207 1.08207i 0.0857552 0.996316i \(-0.472670\pi\)
0.996316 0.0857552i \(-0.0273303\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6109.40i 0.419884i
\(597\) 0 0
\(598\) 756.000 + 756.000i 0.0516976 + 0.0516976i
\(599\) 9011.37 0.614682 0.307341 0.951599i \(-0.400561\pi\)
0.307341 + 0.951599i \(0.400561\pi\)
\(600\) 0 0
\(601\) 15388.0 1.04441 0.522204 0.852820i \(-0.325110\pi\)
0.522204 + 0.852820i \(0.325110\pi\)
\(602\) −9164.10 9164.10i −0.620434 0.620434i
\(603\) 0 0
\(604\) 512.000i 0.0344917i
\(605\) 0 0
\(606\) 0 0
\(607\) 3177.00 3177.00i 0.212439 0.212439i −0.592864 0.805303i \(-0.702003\pi\)
0.805303 + 0.592864i \(0.202003\pi\)
\(608\) 859.842 859.842i 0.0573539 0.0573539i
\(609\) 0 0
\(610\) 0 0
\(611\) 4582.05i 0.303388i
\(612\) 0 0
\(613\) −1701.00 1701.00i −0.112076 0.112076i 0.648845 0.760921i \(-0.275253\pi\)
−0.760921 + 0.648845i \(0.775253\pi\)
\(614\) 6109.40 0.401556
\(615\) 0 0
\(616\) 3888.00 0.254305
\(617\) −18782.2 18782.2i −1.22551 1.22551i −0.965644 0.259870i \(-0.916320\pi\)
−0.259870 0.965644i \(-0.583680\pi\)
\(618\) 0 0
\(619\) 12548.0i 0.814777i 0.913255 + 0.407388i \(0.133560\pi\)
−0.913255 + 0.407388i \(0.866440\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 6588.00 6588.00i 0.424686 0.424686i
\(623\) 13402.5 13402.5i 0.861894 0.861894i
\(624\) 0 0
\(625\) 0 0
\(626\) 16240.8i 1.03692i
\(627\) 0 0
\(628\) 6732.00 + 6732.00i 0.427764 + 0.427764i
\(629\) −30776.1 −1.95091
\(630\) 0 0
\(631\) −27340.0 −1.72486 −0.862431 0.506174i \(-0.831059\pi\)
−0.862431 + 0.506174i \(0.831059\pi\)
\(632\) 1538.66 + 1538.66i 0.0968430 + 0.0968430i
\(633\) 0 0
\(634\) 11832.0i 0.741181i
\(635\) 0 0
\(636\) 0 0
\(637\) −1629.00 + 1629.00i −0.101324 + 0.101324i
\(638\) −4123.85 + 4123.85i −0.255901 + 0.255901i
\(639\) 0 0
\(640\) 0 0
\(641\) 21039.3i 1.29641i 0.761465 + 0.648206i \(0.224481\pi\)
−0.761465 + 0.648206i \(0.775519\pi\)
\(642\) 0 0
\(643\) 11394.0 + 11394.0i 0.698811 + 0.698811i 0.964154 0.265343i \(-0.0854852\pi\)
−0.265343 + 0.964154i \(0.585485\pi\)
\(644\) 2138.29 0.130839
\(645\) 0 0
\(646\) −5928.00 −0.361043
\(647\) −6584.58 6584.58i −0.400103 0.400103i 0.478166 0.878269i \(-0.341302\pi\)
−0.878269 + 0.478166i \(0.841302\pi\)
\(648\) 0 0
\(649\) 13122.0i 0.793657i
\(650\) 0 0
\(651\) 0 0
\(652\) −792.000 + 792.000i −0.0475723 + 0.0475723i
\(653\) −7008.84 + 7008.84i −0.420026 + 0.420026i −0.885213 0.465186i \(-0.845987\pi\)
0.465186 + 0.885213i \(0.345987\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 5498.46i 0.327254i
\(657\) 0 0
\(658\) 6480.00 + 6480.00i 0.383916 + 0.383916i
\(659\) −32112.5 −1.89822 −0.949110 0.314944i \(-0.898014\pi\)
−0.949110 + 0.314944i \(0.898014\pi\)
\(660\) 0 0
\(661\) 31822.0 1.87251 0.936257 0.351315i \(-0.114265\pi\)
0.936257 + 0.351315i \(0.114265\pi\)
\(662\) −3006.62 3006.62i −0.176519 0.176519i
\(663\) 0 0
\(664\) 672.000i 0.0392751i
\(665\) 0 0
\(666\) 0 0
\(667\) −2268.00 + 2268.00i −0.131660 + 0.131660i
\(668\) 11930.3 11930.3i 0.691014 0.691014i
\(669\) 0 0
\(670\) 0 0
\(671\) 4200.21i 0.241650i
\(672\) 0 0
\(673\) 3330.00 + 3330.00i 0.190731 + 0.190731i 0.796012 0.605281i \(-0.206939\pi\)
−0.605281 + 0.796012i \(0.706939\pi\)
\(674\) 16851.8 0.963065
\(675\) 0 0
\(676\) 8140.00 0.463132
\(677\) −5600.29 5600.29i −0.317927 0.317927i 0.530044 0.847970i \(-0.322176\pi\)
−0.847970 + 0.530044i \(0.822176\pi\)
\(678\) 0 0
\(679\) 12312.0i 0.695863i
\(680\) 0 0
\(681\) 0 0
\(682\) 12744.0 12744.0i 0.715532 0.715532i
\(683\) 17581.5 17581.5i 0.984974 0.984974i −0.0149144 0.999889i \(-0.504748\pi\)
0.999889 + 0.0149144i \(0.00474757\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 13338.9i 0.742391i
\(687\) 0 0
\(688\) 5760.00 + 5760.00i 0.319183 + 0.319183i
\(689\) −6338.51 −0.350476
\(690\) 0 0
\(691\) −23330.0 −1.28439 −0.642196 0.766540i \(-0.721977\pi\)
−0.642196 + 0.766540i \(0.721977\pi\)
\(692\) 6737.31 + 6737.31i 0.370107 + 0.370107i
\(693\) 0 0
\(694\) 5976.00i 0.326867i
\(695\) 0 0
\(696\) 0 0
\(697\) 18954.0 18954.0i 1.03003 1.03003i
\(698\) −3668.47 + 3668.47i −0.198931 + 0.198931i
\(699\) 0 0
\(700\) 0 0
\(701\) 8018.59i 0.432037i 0.976389 + 0.216019i \(0.0693072\pi\)
−0.976389 + 0.216019i \(0.930693\pi\)
\(702\) 0 0
\(703\) 10602.0 + 10602.0i 0.568794 + 0.568794i
\(704\) −2443.76 −0.130828
\(705\) 0 0
\(706\) −2772.00 −0.147770
\(707\) −7560.39 7560.39i −0.402175 0.402175i
\(708\) 0 0
\(709\) 25414.0i 1.34618i 0.739560 + 0.673091i \(0.235034\pi\)
−0.739560 + 0.673091i \(0.764966\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −8424.00 + 8424.00i −0.443403 + 0.443403i
\(713\) 7008.84 7008.84i 0.368139 0.368139i
\(714\) 0 0
\(715\) 0 0
\(716\) 1374.62i 0.0717483i
\(717\) 0 0
\(718\) −12744.0 12744.0i −0.662398 0.662398i
\(719\) 10386.0 0.538709 0.269355 0.963041i \(-0.413190\pi\)
0.269355 + 0.963041i \(0.413190\pi\)
\(720\) 0 0
\(721\) −1458.00 −0.0753103
\(722\) −7657.97 7657.97i −0.394737 0.394737i
\(723\) 0 0
\(724\) 10088.0i 0.517842i
\(725\) 0 0
\(726\) 0 0
\(727\) 22707.0 22707.0i 1.15840 1.15840i 0.173579 0.984820i \(-0.444467\pi\)
0.984820 0.173579i \(-0.0555332\pi\)
\(728\) −916.410 + 916.410i −0.0466544 + 0.0466544i
\(729\) 0 0
\(730\) 0 0
\(731\) 39711.1i 2.00926i
\(732\) 0 0
\(733\) 12573.0 + 12573.0i 0.633553 + 0.633553i 0.948957 0.315404i \(-0.102140\pi\)
−0.315404 + 0.948957i \(0.602140\pi\)
\(734\) 16266.3 0.817983
\(735\) 0 0
\(736\) −1344.00 −0.0673105
\(737\) 26805.0 + 26805.0i 1.33972 + 1.33972i
\(738\) 0 0
\(739\) 14402.0i 0.716896i −0.933550 0.358448i \(-0.883306\pi\)
0.933550 0.358448i \(-0.116694\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −8964.00 + 8964.00i −0.443502 + 0.443502i
\(743\) −22146.6 + 22146.6i −1.09351 + 1.09351i −0.0983612 + 0.995151i \(0.531360\pi\)
−0.995151 + 0.0983612i \(0.968640\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 6236.68i 0.306087i
\(747\) 0 0
\(748\) 8424.00 + 8424.00i 0.411781 + 0.411781i
\(749\) −18328.2 −0.894123
\(750\) 0 0
\(751\) 15208.0 0.738945 0.369473 0.929242i \(-0.379538\pi\)
0.369473 + 0.929242i \(0.379538\pi\)
\(752\) −4072.94 4072.94i −0.197506 0.197506i
\(753\) 0 0
\(754\) 1944.00i 0.0938943i
\(755\) 0 0
\(756\) 0 0
\(757\) −1485.00 + 1485.00i −0.0712989 + 0.0712989i −0.741857 0.670558i \(-0.766055\pi\)
0.670558 + 0.741857i \(0.266055\pi\)
\(758\) −1046.52 + 1046.52i −0.0501468 + 0.0501468i
\(759\) 0 0
\(760\) 0 0
\(761\) 2176.47i 0.103676i −0.998656 0.0518378i \(-0.983492\pi\)
0.998656 0.0518378i \(-0.0165079\pi\)
\(762\) 0 0
\(763\) −14922.0 14922.0i −0.708011 0.708011i
\(764\) 18022.7 0.853455
\(765\) 0 0
\(766\) 23424.0 1.10489
\(767\) −3092.89 3092.89i −0.145603 0.145603i
\(768\) 0 0
\(769\) 5960.00i 0.279484i 0.990188 + 0.139742i \(0.0446273\pi\)
−0.990188 + 0.139742i \(0.955373\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −5832.00 + 5832.00i −0.271889 + 0.271889i
\(773\) 14569.2 14569.2i 0.677903 0.677903i −0.281623 0.959525i \(-0.590873\pi\)
0.959525 + 0.281623i \(0.0908726\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 7738.58i 0.357988i
\(777\) 0 0
\(778\) −8640.00 8640.00i −0.398148 0.398148i
\(779\) −13058.8 −0.600618
\(780\) 0 0
\(781\) −2916.00 −0.133601
\(782\) 4632.96 + 4632.96i 0.211860 + 0.211860i
\(783\) 0 0
\(784\) 2896.00i 0.131924i
\(785\) 0 0
\(786\) 0 0
\(787\) −21330.0 + 21330.0i −0.966115 + 0.966115i −0.999444 0.0333296i \(-0.989389\pi\)
0.0333296 + 0.999444i \(0.489389\pi\)
\(788\) −441.235 + 441.235i −0.0199471 + 0.0199471i
\(789\) 0 0
\(790\) 0 0
\(791\) 8476.80i 0.381037i
\(792\) 0 0
\(793\) 990.000 + 990.000i 0.0443328 + 0.0443328i
\(794\) 19371.9 0.865847
\(795\) 0 0
\(796\) 2800.00 0.124678
\(797\) −21107.1 21107.1i −0.938084 0.938084i 0.0601076 0.998192i \(-0.480856\pi\)
−0.998192 + 0.0601076i \(0.980856\pi\)
\(798\) 0 0
\(799\) 28080.0i 1.24330i
\(800\) 0 0
\(801\) 0 0
\(802\) 16578.0 16578.0i 0.729912 0.729912i
\(803\) −14433.5 + 14433.5i −0.634304 + 0.634304i
\(804\) 0 0
\(805\) 0 0
\(806\) 6007.58i 0.262541i
\(807\) 0 0
\(808\) 4752.00 + 4752.00i 0.206899 + 0.206899i
\(809\) −3321.99 −0.144369 −0.0721847 0.997391i \(-0.522997\pi\)
−0.0721847 + 0.997391i \(0.522997\pi\)
\(810\) 0 0
\(811\) −27380.0 −1.18550 −0.592751 0.805386i \(-0.701958\pi\)
−0.592751 + 0.805386i \(0.701958\pi\)
\(812\) −2749.23 2749.23i −0.118817 0.118817i
\(813\) 0 0
\(814\) 30132.0i 1.29745i
\(815\) 0 0
\(816\) 0 0
\(817\) −13680.0 + 13680.0i −0.585805 + 0.585805i
\(818\) 6157.49 6157.49i 0.263192 0.263192i
\(819\) 0 0
\(820\) 0 0
\(821\) 14586.2i 0.620051i −0.950728 0.310025i \(-0.899662\pi\)
0.950728 0.310025i \(-0.100338\pi\)
\(822\) 0 0
\(823\) −11079.0 11079.0i −0.469246 0.469246i 0.432424 0.901670i \(-0.357658\pi\)
−0.901670 + 0.432424i \(0.857658\pi\)
\(824\) 916.410 0.0387435
\(825\) 0 0
\(826\) −8748.00 −0.368501
\(827\) 12286.7 + 12286.7i 0.516626 + 0.516626i 0.916549 0.399923i \(-0.130963\pi\)
−0.399923 + 0.916549i \(0.630963\pi\)
\(828\) 0 0
\(829\) 12854.0i 0.538526i −0.963067 0.269263i \(-0.913220\pi\)
0.963067 0.269263i \(-0.0867801\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 576.000 576.000i 0.0240015 0.0240015i
\(833\) −9982.93 + 9982.93i −0.415232 + 0.415232i
\(834\) 0 0
\(835\) 0 0
\(836\) 5803.93i 0.240111i
\(837\) 0 0
\(838\) −19386.0 19386.0i −0.799139 0.799139i
\(839\) 18480.9 0.760468 0.380234 0.924890i \(-0.375843\pi\)
0.380234 + 0.924890i \(0.375843\pi\)
\(840\) 0 0
\(841\) −18557.0 −0.760876
\(842\) −12063.2 12063.2i −0.493737 0.493737i
\(843\) 0 0
\(844\) 16336.0i 0.666242i
\(845\) 0 0
\(846\) 0 0
\(847\) 1143.00 1143.00i 0.0463683 0.0463683i
\(848\) 5634.23 5634.23i 0.228161 0.228161i
\(849\) 0 0
\(850\) 0 0
\(851\) 16571.8i 0.667535i
\(852\) 0 0
\(853\) −28377.0 28377.0i −1.13905 1.13905i −0.988621 0.150430i \(-0.951934\pi\)
−0.150430 0.988621i \(-0.548066\pi\)
\(854\) 2800.14 0.112200
\(855\) 0 0
\(856\) 11520.0 0.459983
\(857\) 19732.5 + 19732.5i 0.786523 + 0.786523i 0.980922 0.194400i \(-0.0622758\pi\)
−0.194400 + 0.980922i \(0.562276\pi\)
\(858\) 0 0
\(859\) 17084.0i 0.678578i 0.940682 + 0.339289i \(0.110186\pi\)
−0.940682 + 0.339289i \(0.889814\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −19224.0 + 19224.0i −0.759596 + 0.759596i
\(863\) −17280.3 + 17280.3i −0.681608 + 0.681608i −0.960362 0.278755i \(-0.910078\pi\)
0.278755 + 0.960362i \(0.410078\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 13084.3i 0.513421i
\(867\) 0 0
\(868\) 8496.00 + 8496.00i 0.332227 + 0.332227i
\(869\) 10386.0 0.405432
\(870\) 0 0
\(871\) −12636.0 −0.491567
\(872\) 9379.06 + 9379.06i 0.364238 + 0.364238i
\(873\) 0 0
\(874\) 3192.00i 0.123537i
\(875\) 0 0
\(876\) 0 0
\(877\) −14805.0 + 14805.0i −0.570045 + 0.570045i −0.932141 0.362096i \(-0.882061\pi\)
0.362096 + 0.932141i \(0.382061\pi\)
\(878\) −19929.1 + 19929.1i −0.766030 + 0.766030i
\(879\) 0 0
\(880\) 0 0
\(881\) 23635.8i 0.903869i −0.892051 0.451935i \(-0.850734\pi\)
0.892051 0.451935i \(-0.149266\pi\)
\(882\) 0 0
\(883\) −5958.00 5958.00i −0.227070 0.227070i 0.584398 0.811467i \(-0.301331\pi\)
−0.811467 + 0.584398i \(0.801331\pi\)
\(884\) −3971.11 −0.151089
\(885\) 0 0
\(886\) −7536.00 −0.285753
\(887\) 13164.9 + 13164.9i 0.498348 + 0.498348i 0.910923 0.412576i \(-0.135371\pi\)
−0.412576 + 0.910923i \(0.635371\pi\)
\(888\) 0 0
\(889\) 13446.0i 0.507272i
\(890\) 0 0
\(891\) 0 0
\(892\) −10548.0 + 10548.0i −0.395934 + 0.395934i
\(893\) 9673.22 9673.22i 0.362488 0.362488i
\(894\) 0 0
\(895\) 0 0
\(896\) 1629.17i 0.0607443i
\(897\) 0 0
\(898\) 1782.00 + 1782.00i 0.0662206 + 0.0662206i
\(899\) −18022.7 −0.668623
\(900\) 0 0
\(901\) −38844.0 −1.43627
\(902\) 18557.3 + 18557.3i 0.685023 + 0.685023i
\(903\) 0 0
\(904\) 5328.00i 0.196025i
\(905\) 0 0
\(906\) 0 0
\(907\) −11088.0 + 11088.0i −0.405922 + 0.405922i −0.880314 0.474392i \(-0.842668\pi\)
0.474392 + 0.880314i \(0.342668\pi\)
\(908\) 6686.40 6686.40i 0.244379 0.244379i
\(909\) 0 0
\(910\) 0 0
\(911\) 3207.44i 0.116649i 0.998298 + 0.0583244i \(0.0185758\pi\)
−0.998298 + 0.0583244i \(0.981424\pi\)
\(912\) 0 0
\(913\) −2268.00 2268.00i −0.0822123 0.0822123i
\(914\) 23419.4 0.847532
\(915\) 0 0
\(916\) −21736.0 −0.784037
\(917\) 15464.4 + 15464.4i 0.556903 + 0.556903i
\(918\) 0 0
\(919\) 46820.0i 1.68058i −0.542140 0.840288i \(-0.682386\pi\)
0.542140 0.840288i \(-0.317614\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 17496.0 17496.0i 0.624946 0.624946i
\(923\) 687.308 687.308i 0.0245103 0.0245103i
\(924\) 0 0
\(925\) 0 0
\(926\) 25023.1i 0.888024i
\(927\) 0 0
\(928\) 1728.00 + 1728.00i 0.0611254 + 0.0611254i
\(929\) −44331.4 −1.56562 −0.782812 0.622258i \(-0.786215\pi\)
−0.782812 + 0.622258i \(0.786215\pi\)
\(930\) 0 0
\(931\) 6878.00 0.242124
\(932\) 16139.0 + 16139.0i 0.567222 + 0.567222i
\(933\) 0 0
\(934\) 13992.0i 0.490184i
\(935\) 0 0
\(936\) 0 0
\(937\) −23688.0 + 23688.0i −0.825884 + 0.825884i −0.986945 0.161061i \(-0.948509\pi\)
0.161061 + 0.986945i \(0.448509\pi\)
\(938\) −17870.0 + 17870.0i −0.622043 + 0.622043i
\(939\) 0 0
\(940\) 0 0
\(941\) 17259.1i 0.597906i 0.954268 + 0.298953i \(0.0966373\pi\)
−0.954268 + 0.298953i \(0.903363\pi\)
\(942\) 0 0
\(943\) 10206.0 + 10206.0i 0.352442 + 0.352442i
\(944\) 5498.46 0.189576
\(945\) 0 0
\(946\) 38880.0 1.33626
\(947\) −441.235 441.235i −0.0151407 0.0151407i 0.699496 0.714637i \(-0.253408\pi\)
−0.714637 + 0.699496i \(0.753408\pi\)
\(948\) 0 0
\(949\) 6804.00i 0.232737i
\(950\) 0 0
\(951\) 0 0
\(952\) −5616.00 + 5616.00i −0.191193 + 0.191193i
\(953\) −29414.2 + 29414.2i −0.999811 + 0.999811i −1.00000 0.000188542i \(-0.999940\pi\)
0.000188542 1.00000i \(0.499940\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 14357.1i 0.485713i
\(957\) 0 0
\(958\) −18252.0 18252.0i −0.615548 0.615548i
\(959\) 17335.4 0.583723
\(960\) 0 0
\(961\) 25905.0 0.869558
\(962\) 7102.18 + 7102.18i 0.238029 + 0.238029i
\(963\) 0 0
\(964\) 24848.0i 0.830187i
\(965\) 0 0
\(966\) 0 0
\(967\) −33075.0 + 33075.0i −1.09992 + 1.09992i −0.105498 + 0.994420i \(0.533644\pi\)
−0.994420 + 0.105498i \(0.966356\pi\)
\(968\) −718.420 + 718.420i −0.0238542 + 0.0238542i
\(969\) 0 0
\(970\) 0 0
\(971\) 35396.4i 1.16985i −0.811088 0.584924i \(-0.801124\pi\)
0.811088 0.584924i \(-0.198876\pi\)
\(972\) 0 0
\(973\) −14724.0 14724.0i −0.485128 0.485128i
\(974\) 1043.69 0.0343347
\(975\) 0 0
\(976\) −1760.00 −0.0577215
\(977\) −2711.05 2711.05i −0.0887759 0.0887759i 0.661324 0.750100i \(-0.269995\pi\)
−0.750100 + 0.661324i \(0.769995\pi\)
\(978\) 0 0
\(979\) 56862.0i 1.85630i
\(980\) 0 0
\(981\) 0 0
\(982\) −28026.0 + 28026.0i −0.910739 + 0.910739i
\(983\) 38166.8 38166.8i 1.23838 1.23838i 0.277723 0.960661i \(-0.410420\pi\)
0.960661 0.277723i \(-0.0895798\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 11913.3i 0.384785i
\(987\) 0 0
\(988\) 1368.00 + 1368.00i 0.0440505 + 0.0440505i
\(989\) 21382.9 0.687499
\(990\) 0 0
\(991\) 27308.0 0.875345 0.437673 0.899134i \(-0.355803\pi\)
0.437673 + 0.899134i \(0.355803\pi\)
\(992\) −5340.07 5340.07i −0.170915 0.170915i
\(993\) 0 0
\(994\) 1944.00i 0.0620321i
\(995\) 0 0
\(996\) 0 0
\(997\) −14391.0 + 14391.0i −0.457139 + 0.457139i −0.897715 0.440576i \(-0.854774\pi\)
0.440576 + 0.897715i \(0.354774\pi\)
\(998\) −4324.67 + 4324.67i −0.137169 + 0.137169i
\(999\) 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 450.4.f.a.143.1 yes 4
3.2 odd 2 inner 450.4.f.a.143.2 yes 4
5.2 odd 4 inner 450.4.f.a.107.2 yes 4
5.3 odd 4 450.4.f.c.107.1 yes 4
5.4 even 2 450.4.f.c.143.2 yes 4
15.2 even 4 inner 450.4.f.a.107.1 4
15.8 even 4 450.4.f.c.107.2 yes 4
15.14 odd 2 450.4.f.c.143.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
450.4.f.a.107.1 4 15.2 even 4 inner
450.4.f.a.107.2 yes 4 5.2 odd 4 inner
450.4.f.a.143.1 yes 4 1.1 even 1 trivial
450.4.f.a.143.2 yes 4 3.2 odd 2 inner
450.4.f.c.107.1 yes 4 5.3 odd 4
450.4.f.c.107.2 yes 4 15.8 even 4
450.4.f.c.143.1 yes 4 15.14 odd 2
450.4.f.c.143.2 yes 4 5.4 even 2