Properties

Label 60.5.k.a.13.3
Level $60$
Weight $5$
Character 60.13
Analytic conductor $6.202$
Analytic rank $0$
Dimension $8$
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] = [60,5,Mod(13,60)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(60, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("60.13");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 60.k (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: \(6.20219778503\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 8x^{6} + 28x^{5} + 97x^{4} - 168x^{3} + 288x^{2} + 864x + 1296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{2}\cdot 5^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 13.3
Root \(3.17086 + 3.17086i\) of defining polynomial
Character \(\chi\) \(=\) 60.13
Dual form 60.5.k.a.37.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(3.67423 + 3.67423i) q^{3} +(-21.4961 + 12.7640i) q^{5} +(-29.2390 + 29.2390i) q^{7} +27.0000i q^{9} +O(q^{10})\) \(q+(3.67423 + 3.67423i) q^{3} +(-21.4961 + 12.7640i) q^{5} +(-29.2390 + 29.2390i) q^{7} +27.0000i q^{9} -100.806 q^{11} +(65.5718 + 65.5718i) q^{13} +(-125.879 - 32.0836i) q^{15} +(-206.336 + 206.336i) q^{17} +260.194i q^{19} -214.862 q^{21} +(210.067 + 210.067i) q^{23} +(299.160 - 548.751i) q^{25} +(-99.2043 + 99.2043i) q^{27} -503.161i q^{29} +1252.33 q^{31} +(-370.386 - 370.386i) q^{33} +(255.317 - 1001.73i) q^{35} +(1812.07 - 1812.07i) q^{37} +481.852i q^{39} -3129.49 q^{41} +(2140.10 + 2140.10i) q^{43} +(-344.628 - 580.393i) q^{45} +(838.104 - 838.104i) q^{47} +691.158i q^{49} -1516.26 q^{51} +(883.512 + 883.512i) q^{53} +(2166.94 - 1286.69i) q^{55} +(-956.012 + 956.012i) q^{57} +4933.68i q^{59} -5873.12 q^{61} +(-789.454 - 789.454i) q^{63} +(-2246.49 - 572.576i) q^{65} +(-932.560 + 932.560i) q^{67} +1543.67i q^{69} -4439.96 q^{71} +(-412.029 - 412.029i) q^{73} +(3115.43 - 917.056i) q^{75} +(2947.48 - 2947.48i) q^{77} +8658.90i q^{79} -729.000 q^{81} +(5223.10 + 5223.10i) q^{83} +(1801.74 - 7069.09i) q^{85} +(1848.73 - 1848.73i) q^{87} -5102.20i q^{89} -3834.51 q^{91} +(4601.37 + 4601.37i) q^{93} +(-3321.11 - 5593.13i) q^{95} +(-2418.77 + 2418.77i) q^{97} -2721.77i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{5} - 140 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{5} - 140 q^{7} + 288 q^{11} + 300 q^{13} - 144 q^{15} - 1020 q^{17} + 792 q^{21} + 1320 q^{23} - 2036 q^{25} + 1472 q^{31} - 180 q^{33} + 1416 q^{35} - 300 q^{37} - 3480 q^{41} - 6360 q^{43} + 648 q^{45} + 4800 q^{47} + 2232 q^{51} + 3900 q^{53} + 11172 q^{55} + 360 q^{57} - 11544 q^{61} - 3780 q^{63} - 16380 q^{65} - 920 q^{67} - 3600 q^{71} + 2960 q^{73} + 15912 q^{75} + 19800 q^{77} - 5832 q^{81} + 12720 q^{83} + 1396 q^{85} - 19620 q^{87} + 32400 q^{91} - 14760 q^{93} - 37200 q^{95} - 15600 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(41\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.67423 + 3.67423i 0.408248 + 0.408248i
\(4\) 0 0
\(5\) −21.4961 + 12.7640i −0.859842 + 0.510560i
\(6\) 0 0
\(7\) −29.2390 + 29.2390i −0.596715 + 0.596715i −0.939437 0.342722i \(-0.888651\pi\)
0.342722 + 0.939437i \(0.388651\pi\)
\(8\) 0 0
\(9\) 27.0000i 0.333333i
\(10\) 0 0
\(11\) −100.806 −0.833109 −0.416555 0.909111i \(-0.636763\pi\)
−0.416555 + 0.909111i \(0.636763\pi\)
\(12\) 0 0
\(13\) 65.5718 + 65.5718i 0.387999 + 0.387999i 0.873973 0.485974i \(-0.161535\pi\)
−0.485974 + 0.873973i \(0.661535\pi\)
\(14\) 0 0
\(15\) −125.879 32.0836i −0.559464 0.142594i
\(16\) 0 0
\(17\) −206.336 + 206.336i −0.713967 + 0.713967i −0.967363 0.253396i \(-0.918452\pi\)
0.253396 + 0.967363i \(0.418452\pi\)
\(18\) 0 0
\(19\) 260.194i 0.720758i 0.932806 + 0.360379i \(0.117353\pi\)
−0.932806 + 0.360379i \(0.882647\pi\)
\(20\) 0 0
\(21\) −214.862 −0.487216
\(22\) 0 0
\(23\) 210.067 + 210.067i 0.397101 + 0.397101i 0.877209 0.480108i \(-0.159403\pi\)
−0.480108 + 0.877209i \(0.659403\pi\)
\(24\) 0 0
\(25\) 299.160 548.751i 0.478657 0.878002i
\(26\) 0 0
\(27\) −99.2043 + 99.2043i −0.136083 + 0.136083i
\(28\) 0 0
\(29\) 503.161i 0.598289i −0.954208 0.299145i \(-0.903299\pi\)
0.954208 0.299145i \(-0.0967013\pi\)
\(30\) 0 0
\(31\) 1252.33 1.30316 0.651578 0.758581i \(-0.274107\pi\)
0.651578 + 0.758581i \(0.274107\pi\)
\(32\) 0 0
\(33\) −370.386 370.386i −0.340115 0.340115i
\(34\) 0 0
\(35\) 255.317 1001.73i 0.208422 0.817739i
\(36\) 0 0
\(37\) 1812.07 1812.07i 1.32364 1.32364i 0.412838 0.910805i \(-0.364538\pi\)
0.910805 0.412838i \(-0.135462\pi\)
\(38\) 0 0
\(39\) 481.852i 0.316800i
\(40\) 0 0
\(41\) −3129.49 −1.86168 −0.930842 0.365423i \(-0.880924\pi\)
−0.930842 + 0.365423i \(0.880924\pi\)
\(42\) 0 0
\(43\) 2140.10 + 2140.10i 1.15744 + 1.15744i 0.985025 + 0.172413i \(0.0551563\pi\)
0.172413 + 0.985025i \(0.444844\pi\)
\(44\) 0 0
\(45\) −344.628 580.393i −0.170187 0.286614i
\(46\) 0 0
\(47\) 838.104 838.104i 0.379404 0.379404i −0.491483 0.870887i \(-0.663545\pi\)
0.870887 + 0.491483i \(0.163545\pi\)
\(48\) 0 0
\(49\) 691.158i 0.287863i
\(50\) 0 0
\(51\) −1516.26 −0.582951
\(52\) 0 0
\(53\) 883.512 + 883.512i 0.314529 + 0.314529i 0.846661 0.532132i \(-0.178609\pi\)
−0.532132 + 0.846661i \(0.678609\pi\)
\(54\) 0 0
\(55\) 2166.94 1286.69i 0.716342 0.425352i
\(56\) 0 0
\(57\) −956.012 + 956.012i −0.294248 + 0.294248i
\(58\) 0 0
\(59\) 4933.68i 1.41732i 0.705551 + 0.708659i \(0.250700\pi\)
−0.705551 + 0.708659i \(0.749300\pi\)
\(60\) 0 0
\(61\) −5873.12 −1.57837 −0.789186 0.614154i \(-0.789497\pi\)
−0.789186 + 0.614154i \(0.789497\pi\)
\(62\) 0 0
\(63\) −789.454 789.454i −0.198905 0.198905i
\(64\) 0 0
\(65\) −2246.49 572.576i −0.531714 0.135521i
\(66\) 0 0
\(67\) −932.560 + 932.560i −0.207743 + 0.207743i −0.803308 0.595564i \(-0.796929\pi\)
0.595564 + 0.803308i \(0.296929\pi\)
\(68\) 0 0
\(69\) 1543.67i 0.324232i
\(70\) 0 0
\(71\) −4439.96 −0.880769 −0.440385 0.897809i \(-0.645158\pi\)
−0.440385 + 0.897809i \(0.645158\pi\)
\(72\) 0 0
\(73\) −412.029 412.029i −0.0773183 0.0773183i 0.667390 0.744708i \(-0.267411\pi\)
−0.744708 + 0.667390i \(0.767411\pi\)
\(74\) 0 0
\(75\) 3115.43 917.056i 0.553854 0.163032i
\(76\) 0 0
\(77\) 2947.48 2947.48i 0.497129 0.497129i
\(78\) 0 0
\(79\) 8658.90i 1.38742i 0.720254 + 0.693711i \(0.244025\pi\)
−0.720254 + 0.693711i \(0.755975\pi\)
\(80\) 0 0
\(81\) −729.000 −0.111111
\(82\) 0 0
\(83\) 5223.10 + 5223.10i 0.758180 + 0.758180i 0.975991 0.217811i \(-0.0698917\pi\)
−0.217811 + 0.975991i \(0.569892\pi\)
\(84\) 0 0
\(85\) 1801.74 7069.09i 0.249376 0.978421i
\(86\) 0 0
\(87\) 1848.73 1848.73i 0.244251 0.244251i
\(88\) 0 0
\(89\) 5102.20i 0.644136i −0.946717 0.322068i \(-0.895622\pi\)
0.946717 0.322068i \(-0.104378\pi\)
\(90\) 0 0
\(91\) −3834.51 −0.463049
\(92\) 0 0
\(93\) 4601.37 + 4601.37i 0.532012 + 0.532012i
\(94\) 0 0
\(95\) −3321.11 5593.13i −0.367990 0.619738i
\(96\) 0 0
\(97\) −2418.77 + 2418.77i −0.257070 + 0.257070i −0.823861 0.566791i \(-0.808185\pi\)
0.566791 + 0.823861i \(0.308185\pi\)
\(98\) 0 0
\(99\) 2721.77i 0.277703i
\(100\) 0 0
\(101\) 4380.82 0.429450 0.214725 0.976675i \(-0.431114\pi\)
0.214725 + 0.976675i \(0.431114\pi\)
\(102\) 0 0
\(103\) −6844.96 6844.96i −0.645203 0.645203i 0.306627 0.951830i \(-0.400800\pi\)
−0.951830 + 0.306627i \(0.900800\pi\)
\(104\) 0 0
\(105\) 4618.69 2742.50i 0.418929 0.248753i
\(106\) 0 0
\(107\) 12713.4 12713.4i 1.11044 1.11044i 0.117350 0.993091i \(-0.462560\pi\)
0.993091 0.117350i \(-0.0374398\pi\)
\(108\) 0 0
\(109\) 11845.5i 0.997011i −0.866887 0.498505i \(-0.833882\pi\)
0.866887 0.498505i \(-0.166118\pi\)
\(110\) 0 0
\(111\) 13315.9 1.08075
\(112\) 0 0
\(113\) 16798.8 + 16798.8i 1.31559 + 1.31559i 0.917226 + 0.398367i \(0.130423\pi\)
0.398367 + 0.917226i \(0.369577\pi\)
\(114\) 0 0
\(115\) −7196.89 1834.31i −0.544188 0.138700i
\(116\) 0 0
\(117\) −1770.44 + 1770.44i −0.129333 + 0.129333i
\(118\) 0 0
\(119\) 12066.1i 0.852069i
\(120\) 0 0
\(121\) −4479.11 −0.305929
\(122\) 0 0
\(123\) −11498.5 11498.5i −0.760029 0.760029i
\(124\) 0 0
\(125\) 573.499 + 15614.5i 0.0367039 + 0.999326i
\(126\) 0 0
\(127\) −14949.7 + 14949.7i −0.926882 + 0.926882i −0.997503 0.0706214i \(-0.977502\pi\)
0.0706214 + 0.997503i \(0.477502\pi\)
\(128\) 0 0
\(129\) 15726.5i 0.945044i
\(130\) 0 0
\(131\) −3761.56 −0.219193 −0.109596 0.993976i \(-0.534956\pi\)
−0.109596 + 0.993976i \(0.534956\pi\)
\(132\) 0 0
\(133\) −7607.81 7607.81i −0.430087 0.430087i
\(134\) 0 0
\(135\) 866.257 3398.75i 0.0475312 0.186488i
\(136\) 0 0
\(137\) −13779.0 + 13779.0i −0.734139 + 0.734139i −0.971437 0.237298i \(-0.923738\pi\)
0.237298 + 0.971437i \(0.423738\pi\)
\(138\) 0 0
\(139\) 25283.6i 1.30861i 0.756232 + 0.654303i \(0.227038\pi\)
−0.756232 + 0.654303i \(0.772962\pi\)
\(140\) 0 0
\(141\) 6158.78 0.309782
\(142\) 0 0
\(143\) −6610.04 6610.04i −0.323245 0.323245i
\(144\) 0 0
\(145\) 6422.36 + 10816.0i 0.305463 + 0.514434i
\(146\) 0 0
\(147\) −2539.48 + 2539.48i −0.117519 + 0.117519i
\(148\) 0 0
\(149\) 13593.6i 0.612295i −0.951984 0.306147i \(-0.900960\pi\)
0.951984 0.306147i \(-0.0990400\pi\)
\(150\) 0 0
\(151\) 17658.0 0.774440 0.387220 0.921987i \(-0.373435\pi\)
0.387220 + 0.921987i \(0.373435\pi\)
\(152\) 0 0
\(153\) −5571.08 5571.08i −0.237989 0.237989i
\(154\) 0 0
\(155\) −26920.2 + 15984.8i −1.12051 + 0.665340i
\(156\) 0 0
\(157\) 2986.67 2986.67i 0.121168 0.121168i −0.643923 0.765091i \(-0.722694\pi\)
0.765091 + 0.643923i \(0.222694\pi\)
\(158\) 0 0
\(159\) 6492.46i 0.256812i
\(160\) 0 0
\(161\) −12284.3 −0.473913
\(162\) 0 0
\(163\) −14591.9 14591.9i −0.549207 0.549207i 0.377005 0.926211i \(-0.376954\pi\)
−0.926211 + 0.377005i \(0.876954\pi\)
\(164\) 0 0
\(165\) 12689.4 + 3234.22i 0.466095 + 0.118796i
\(166\) 0 0
\(167\) 4074.21 4074.21i 0.146087 0.146087i −0.630281 0.776367i \(-0.717060\pi\)
0.776367 + 0.630281i \(0.217060\pi\)
\(168\) 0 0
\(169\) 19961.7i 0.698914i
\(170\) 0 0
\(171\) −7025.23 −0.240253
\(172\) 0 0
\(173\) 10988.5 + 10988.5i 0.367152 + 0.367152i 0.866437 0.499286i \(-0.166404\pi\)
−0.499286 + 0.866437i \(0.666404\pi\)
\(174\) 0 0
\(175\) 7297.80 + 24792.1i 0.238295 + 0.809539i
\(176\) 0 0
\(177\) −18127.5 + 18127.5i −0.578618 + 0.578618i
\(178\) 0 0
\(179\) 55895.4i 1.74450i −0.489063 0.872248i \(-0.662661\pi\)
0.489063 0.872248i \(-0.337339\pi\)
\(180\) 0 0
\(181\) 61470.9 1.87634 0.938171 0.346171i \(-0.112518\pi\)
0.938171 + 0.346171i \(0.112518\pi\)
\(182\) 0 0
\(183\) −21579.2 21579.2i −0.644368 0.644368i
\(184\) 0 0
\(185\) −15823.0 + 62081.5i −0.462324 + 1.81392i
\(186\) 0 0
\(187\) 20800.0 20800.0i 0.594812 0.594812i
\(188\) 0 0
\(189\) 5801.28i 0.162405i
\(190\) 0 0
\(191\) 21064.5 0.577411 0.288706 0.957418i \(-0.406775\pi\)
0.288706 + 0.957418i \(0.406775\pi\)
\(192\) 0 0
\(193\) −16416.6 16416.6i −0.440726 0.440726i 0.451530 0.892256i \(-0.350878\pi\)
−0.892256 + 0.451530i \(0.850878\pi\)
\(194\) 0 0
\(195\) −6150.36 10357.9i −0.161745 0.272398i
\(196\) 0 0
\(197\) 37084.2 37084.2i 0.955556 0.955556i −0.0434975 0.999054i \(-0.513850\pi\)
0.999054 + 0.0434975i \(0.0138500\pi\)
\(198\) 0 0
\(199\) 15577.7i 0.393366i 0.980467 + 0.196683i \(0.0630170\pi\)
−0.980467 + 0.196683i \(0.936983\pi\)
\(200\) 0 0
\(201\) −6852.89 −0.169622
\(202\) 0 0
\(203\) 14712.0 + 14712.0i 0.357008 + 0.357008i
\(204\) 0 0
\(205\) 67271.7 39944.8i 1.60075 0.950501i
\(206\) 0 0
\(207\) −5671.80 + 5671.80i −0.132367 + 0.132367i
\(208\) 0 0
\(209\) 26229.1i 0.600470i
\(210\) 0 0
\(211\) 71670.6 1.60981 0.804907 0.593401i \(-0.202215\pi\)
0.804907 + 0.593401i \(0.202215\pi\)
\(212\) 0 0
\(213\) −16313.4 16313.4i −0.359573 0.359573i
\(214\) 0 0
\(215\) −73320.0 18687.5i −1.58615 0.404272i
\(216\) 0 0
\(217\) −36617.0 + 36617.0i −0.777613 + 0.777613i
\(218\) 0 0
\(219\) 3027.78i 0.0631301i
\(220\) 0 0
\(221\) −27059.7 −0.554036
\(222\) 0 0
\(223\) 10834.2 + 10834.2i 0.217864 + 0.217864i 0.807598 0.589734i \(-0.200767\pi\)
−0.589734 + 0.807598i \(0.700767\pi\)
\(224\) 0 0
\(225\) 14816.3 + 8077.33i 0.292667 + 0.159552i
\(226\) 0 0
\(227\) 8536.10 8536.10i 0.165656 0.165656i −0.619411 0.785067i \(-0.712628\pi\)
0.785067 + 0.619411i \(0.212628\pi\)
\(228\) 0 0
\(229\) 53172.3i 1.01395i −0.861962 0.506973i \(-0.830765\pi\)
0.861962 0.506973i \(-0.169235\pi\)
\(230\) 0 0
\(231\) 21659.4 0.405904
\(232\) 0 0
\(233\) −8276.62 8276.62i −0.152455 0.152455i 0.626759 0.779213i \(-0.284381\pi\)
−0.779213 + 0.626759i \(0.784381\pi\)
\(234\) 0 0
\(235\) −7318.36 + 28713.5i −0.132519 + 0.519936i
\(236\) 0 0
\(237\) −31814.8 + 31814.8i −0.566412 + 0.566412i
\(238\) 0 0
\(239\) 64206.9i 1.12405i 0.827120 + 0.562025i \(0.189978\pi\)
−0.827120 + 0.562025i \(0.810022\pi\)
\(240\) 0 0
\(241\) −52792.1 −0.908940 −0.454470 0.890762i \(-0.650171\pi\)
−0.454470 + 0.890762i \(0.650171\pi\)
\(242\) 0 0
\(243\) −2678.52 2678.52i −0.0453609 0.0453609i
\(244\) 0 0
\(245\) −8821.94 14857.2i −0.146971 0.247516i
\(246\) 0 0
\(247\) −17061.4 + 17061.4i −0.279653 + 0.279653i
\(248\) 0 0
\(249\) 38381.8i 0.619051i
\(250\) 0 0
\(251\) 7619.52 0.120943 0.0604714 0.998170i \(-0.480740\pi\)
0.0604714 + 0.998170i \(0.480740\pi\)
\(252\) 0 0
\(253\) −21176.0 21176.0i −0.330829 0.330829i
\(254\) 0 0
\(255\) 32593.5 19353.5i 0.501246 0.297632i
\(256\) 0 0
\(257\) −56900.1 + 56900.1i −0.861483 + 0.861483i −0.991510 0.130027i \(-0.958494\pi\)
0.130027 + 0.991510i \(0.458494\pi\)
\(258\) 0 0
\(259\) 105966.i 1.57967i
\(260\) 0 0
\(261\) 13585.4 0.199430
\(262\) 0 0
\(263\) 78565.0 + 78565.0i 1.13584 + 1.13584i 0.989188 + 0.146653i \(0.0468502\pi\)
0.146653 + 0.989188i \(0.453150\pi\)
\(264\) 0 0
\(265\) −30269.2 7714.86i −0.431031 0.109859i
\(266\) 0 0
\(267\) 18746.7 18746.7i 0.262967 0.262967i
\(268\) 0 0
\(269\) 108837.i 1.50409i −0.659114 0.752043i \(-0.729068\pi\)
0.659114 0.752043i \(-0.270932\pi\)
\(270\) 0 0
\(271\) −51081.3 −0.695542 −0.347771 0.937580i \(-0.613061\pi\)
−0.347771 + 0.937580i \(0.613061\pi\)
\(272\) 0 0
\(273\) −14088.9 14088.9i −0.189039 0.189039i
\(274\) 0 0
\(275\) −30157.2 + 55317.5i −0.398773 + 0.731472i
\(276\) 0 0
\(277\) −37506.3 + 37506.3i −0.488815 + 0.488815i −0.907932 0.419117i \(-0.862340\pi\)
0.419117 + 0.907932i \(0.362340\pi\)
\(278\) 0 0
\(279\) 33813.0i 0.434386i
\(280\) 0 0
\(281\) −101190. −1.28152 −0.640758 0.767743i \(-0.721380\pi\)
−0.640758 + 0.767743i \(0.721380\pi\)
\(282\) 0 0
\(283\) −4039.24 4039.24i −0.0504344 0.0504344i 0.681440 0.731874i \(-0.261354\pi\)
−0.731874 + 0.681440i \(0.761354\pi\)
\(284\) 0 0
\(285\) 8347.94 32753.0i 0.102776 0.403238i
\(286\) 0 0
\(287\) 91503.2 91503.2i 1.11089 1.11089i
\(288\) 0 0
\(289\) 1628.37i 0.0194966i
\(290\) 0 0
\(291\) −17774.3 −0.209897
\(292\) 0 0
\(293\) 92563.9 + 92563.9i 1.07822 + 1.07822i 0.996669 + 0.0815481i \(0.0259864\pi\)
0.0815481 + 0.996669i \(0.474014\pi\)
\(294\) 0 0
\(295\) −62973.6 106055.i −0.723626 1.21867i
\(296\) 0 0
\(297\) 10000.4 10000.4i 0.113372 0.113372i
\(298\) 0 0
\(299\) 27548.9i 0.308150i
\(300\) 0 0
\(301\) −125149. −1.38132
\(302\) 0 0
\(303\) 16096.2 + 16096.2i 0.175322 + 0.175322i
\(304\) 0 0
\(305\) 126249. 74964.6i 1.35715 0.805854i
\(306\) 0 0
\(307\) 82316.5 82316.5i 0.873394 0.873394i −0.119447 0.992841i \(-0.538112\pi\)
0.992841 + 0.119447i \(0.0381121\pi\)
\(308\) 0 0
\(309\) 50300.0i 0.526806i
\(310\) 0 0
\(311\) 66895.0 0.691628 0.345814 0.938303i \(-0.387603\pi\)
0.345814 + 0.938303i \(0.387603\pi\)
\(312\) 0 0
\(313\) 12902.7 + 12902.7i 0.131702 + 0.131702i 0.769885 0.638183i \(-0.220314\pi\)
−0.638183 + 0.769885i \(0.720314\pi\)
\(314\) 0 0
\(315\) 27046.7 + 6893.55i 0.272580 + 0.0694739i
\(316\) 0 0
\(317\) 87261.9 87261.9i 0.868373 0.868373i −0.123919 0.992292i \(-0.539546\pi\)
0.992292 + 0.123919i \(0.0395464\pi\)
\(318\) 0 0
\(319\) 50721.8i 0.498440i
\(320\) 0 0
\(321\) 93424.2 0.906671
\(322\) 0 0
\(323\) −53687.4 53687.4i −0.514597 0.514597i
\(324\) 0 0
\(325\) 55599.1 16366.1i 0.526382 0.154946i
\(326\) 0 0
\(327\) 43523.1 43523.1i 0.407028 0.407028i
\(328\) 0 0
\(329\) 49010.7i 0.452792i
\(330\) 0 0
\(331\) −89537.0 −0.817234 −0.408617 0.912706i \(-0.633989\pi\)
−0.408617 + 0.912706i \(0.633989\pi\)
\(332\) 0 0
\(333\) 48925.8 + 48925.8i 0.441214 + 0.441214i
\(334\) 0 0
\(335\) 8143.16 31949.6i 0.0725610 0.284692i
\(336\) 0 0
\(337\) −10633.8 + 10633.8i −0.0936326 + 0.0936326i −0.752372 0.658739i \(-0.771090\pi\)
0.658739 + 0.752372i \(0.271090\pi\)
\(338\) 0 0
\(339\) 123446.i 1.07418i
\(340\) 0 0
\(341\) −126243. −1.08567
\(342\) 0 0
\(343\) −90411.7 90411.7i −0.768487 0.768487i
\(344\) 0 0
\(345\) −19703.4 33182.8i −0.165540 0.278788i
\(346\) 0 0
\(347\) −37803.6 + 37803.6i −0.313960 + 0.313960i −0.846442 0.532482i \(-0.821260\pi\)
0.532482 + 0.846442i \(0.321260\pi\)
\(348\) 0 0
\(349\) 68613.6i 0.563325i 0.959514 + 0.281663i \(0.0908859\pi\)
−0.959514 + 0.281663i \(0.909114\pi\)
\(350\) 0 0
\(351\) −13010.0 −0.105600
\(352\) 0 0
\(353\) 31348.2 + 31348.2i 0.251572 + 0.251572i 0.821615 0.570043i \(-0.193073\pi\)
−0.570043 + 0.821615i \(0.693073\pi\)
\(354\) 0 0
\(355\) 95441.6 56671.7i 0.757323 0.449686i
\(356\) 0 0
\(357\) 44333.9 44333.9i 0.347856 0.347856i
\(358\) 0 0
\(359\) 85046.4i 0.659883i −0.944001 0.329941i \(-0.892971\pi\)
0.944001 0.329941i \(-0.107029\pi\)
\(360\) 0 0
\(361\) 62620.3 0.480508
\(362\) 0 0
\(363\) −16457.3 16457.3i −0.124895 0.124895i
\(364\) 0 0
\(365\) 14116.1 + 3597.86i 0.105957 + 0.0270059i
\(366\) 0 0
\(367\) −19597.4 + 19597.4i −0.145501 + 0.145501i −0.776105 0.630604i \(-0.782807\pi\)
0.630604 + 0.776105i \(0.282807\pi\)
\(368\) 0 0
\(369\) 84496.2i 0.620561i
\(370\) 0 0
\(371\) −51666.0 −0.375368
\(372\) 0 0
\(373\) 88837.3 + 88837.3i 0.638525 + 0.638525i 0.950191 0.311667i \(-0.100887\pi\)
−0.311667 + 0.950191i \(0.600887\pi\)
\(374\) 0 0
\(375\) −55264.1 + 59478.4i −0.392989 + 0.422958i
\(376\) 0 0
\(377\) 32993.2 32993.2i 0.232136 0.232136i
\(378\) 0 0
\(379\) 55861.4i 0.388896i −0.980913 0.194448i \(-0.937708\pi\)
0.980913 0.194448i \(-0.0622916\pi\)
\(380\) 0 0
\(381\) −109857. −0.756796
\(382\) 0 0
\(383\) 77604.9 + 77604.9i 0.529044 + 0.529044i 0.920287 0.391244i \(-0.127955\pi\)
−0.391244 + 0.920287i \(0.627955\pi\)
\(384\) 0 0
\(385\) −25737.5 + 100981.i −0.173638 + 0.681266i
\(386\) 0 0
\(387\) −57782.8 + 57782.8i −0.385813 + 0.385813i
\(388\) 0 0
\(389\) 210631.i 1.39195i 0.718067 + 0.695974i \(0.245027\pi\)
−0.718067 + 0.695974i \(0.754973\pi\)
\(390\) 0 0
\(391\) −86688.7 −0.567034
\(392\) 0 0
\(393\) −13820.9 13820.9i −0.0894850 0.0894850i
\(394\) 0 0
\(395\) −110522. 186132.i −0.708362 1.19296i
\(396\) 0 0
\(397\) −47327.7 + 47327.7i −0.300285 + 0.300285i −0.841125 0.540840i \(-0.818106\pi\)
0.540840 + 0.841125i \(0.318106\pi\)
\(398\) 0 0
\(399\) 55905.7i 0.351164i
\(400\) 0 0
\(401\) −16778.8 −0.104345 −0.0521726 0.998638i \(-0.516615\pi\)
−0.0521726 + 0.998638i \(0.516615\pi\)
\(402\) 0 0
\(403\) 82117.7 + 82117.7i 0.505623 + 0.505623i
\(404\) 0 0
\(405\) 15670.6 9304.96i 0.0955380 0.0567289i
\(406\) 0 0
\(407\) −182667. + 182667.i −1.10274 + 1.10274i
\(408\) 0 0
\(409\) 33710.4i 0.201520i 0.994911 + 0.100760i \(0.0321274\pi\)
−0.994911 + 0.100760i \(0.967873\pi\)
\(410\) 0 0
\(411\) −101255. −0.599422
\(412\) 0 0
\(413\) −144256. 144256.i −0.845735 0.845735i
\(414\) 0 0
\(415\) −178944. 45608.4i −1.03901 0.264818i
\(416\) 0 0
\(417\) −92897.9 + 92897.9i −0.534237 + 0.534237i
\(418\) 0 0
\(419\) 293984.i 1.67454i 0.546789 + 0.837270i \(0.315850\pi\)
−0.546789 + 0.837270i \(0.684150\pi\)
\(420\) 0 0
\(421\) 90452.8 0.510338 0.255169 0.966896i \(-0.417869\pi\)
0.255169 + 0.966896i \(0.417869\pi\)
\(422\) 0 0
\(423\) 22628.8 + 22628.8i 0.126468 + 0.126468i
\(424\) 0 0
\(425\) 51499.7 + 174955.i 0.285119 + 0.968609i
\(426\) 0 0
\(427\) 171724. 171724.i 0.941838 0.941838i
\(428\) 0 0
\(429\) 48573.7i 0.263929i
\(430\) 0 0
\(431\) 193225. 1.04018 0.520091 0.854111i \(-0.325898\pi\)
0.520091 + 0.854111i \(0.325898\pi\)
\(432\) 0 0
\(433\) 233890. + 233890.i 1.24749 + 1.24749i 0.956826 + 0.290661i \(0.0938753\pi\)
0.290661 + 0.956826i \(0.406125\pi\)
\(434\) 0 0
\(435\) −16143.2 + 63337.7i −0.0853123 + 0.334722i
\(436\) 0 0
\(437\) −54658.0 + 54658.0i −0.286214 + 0.286214i
\(438\) 0 0
\(439\) 256431.i 1.33058i −0.746583 0.665292i \(-0.768307\pi\)
0.746583 0.665292i \(-0.231693\pi\)
\(440\) 0 0
\(441\) −18661.3 −0.0959542
\(442\) 0 0
\(443\) −271033. 271033.i −1.38107 1.38107i −0.842732 0.538333i \(-0.819054\pi\)
−0.538333 0.842732i \(-0.680946\pi\)
\(444\) 0 0
\(445\) 65124.5 + 109677.i 0.328870 + 0.553855i
\(446\) 0 0
\(447\) 49945.9 49945.9i 0.249968 0.249968i
\(448\) 0 0
\(449\) 346521.i 1.71885i −0.511265 0.859423i \(-0.670823\pi\)
0.511265 0.859423i \(-0.329177\pi\)
\(450\) 0 0
\(451\) 315472. 1.55099
\(452\) 0 0
\(453\) 64879.7 + 64879.7i 0.316164 + 0.316164i
\(454\) 0 0
\(455\) 82426.8 48943.7i 0.398149 0.236414i
\(456\) 0 0
\(457\) 37089.6 37089.6i 0.177590 0.177590i −0.612714 0.790305i \(-0.709922\pi\)
0.790305 + 0.612714i \(0.209922\pi\)
\(458\) 0 0
\(459\) 40938.9i 0.194317i
\(460\) 0 0
\(461\) 178438. 0.839627 0.419813 0.907610i \(-0.362096\pi\)
0.419813 + 0.907610i \(0.362096\pi\)
\(462\) 0 0
\(463\) −137047. 137047.i −0.639303 0.639303i 0.311081 0.950383i \(-0.399309\pi\)
−0.950383 + 0.311081i \(0.899309\pi\)
\(464\) 0 0
\(465\) −157643. 40179.4i −0.729070 0.185822i
\(466\) 0 0
\(467\) −137506. + 137506.i −0.630503 + 0.630503i −0.948194 0.317691i \(-0.897092\pi\)
0.317691 + 0.948194i \(0.397092\pi\)
\(468\) 0 0
\(469\) 54534.3i 0.247927i
\(470\) 0 0
\(471\) 21947.5 0.0989334
\(472\) 0 0
\(473\) −215736. 215736.i −0.964272 0.964272i
\(474\) 0 0
\(475\) 142782. + 77839.6i 0.632827 + 0.344995i
\(476\) 0 0
\(477\) −23854.8 + 23854.8i −0.104843 + 0.104843i
\(478\) 0 0
\(479\) 17228.2i 0.0750879i −0.999295 0.0375439i \(-0.988047\pi\)
0.999295 0.0375439i \(-0.0119534\pi\)
\(480\) 0 0
\(481\) 237641. 1.02714
\(482\) 0 0
\(483\) −45135.4 45135.4i −0.193474 0.193474i
\(484\) 0 0
\(485\) 21120.8 82867.2i 0.0897898 0.352289i
\(486\) 0 0
\(487\) 55195.9 55195.9i 0.232728 0.232728i −0.581102 0.813830i \(-0.697378\pi\)
0.813830 + 0.581102i \(0.197378\pi\)
\(488\) 0 0
\(489\) 107228.i 0.448425i
\(490\) 0 0
\(491\) 99534.3 0.412867 0.206433 0.978461i \(-0.433814\pi\)
0.206433 + 0.978461i \(0.433814\pi\)
\(492\) 0 0
\(493\) 103820. + 103820.i 0.427159 + 0.427159i
\(494\) 0 0
\(495\) 34740.6 + 58507.2i 0.141784 + 0.238781i
\(496\) 0 0
\(497\) 129820. 129820.i 0.525568 0.525568i
\(498\) 0 0
\(499\) 222798.i 0.894769i −0.894342 0.447385i \(-0.852355\pi\)
0.894342 0.447385i \(-0.147645\pi\)
\(500\) 0 0
\(501\) 29939.2 0.119279
\(502\) 0 0
\(503\) −18900.9 18900.9i −0.0747045 0.0747045i 0.668767 0.743472i \(-0.266822\pi\)
−0.743472 + 0.668767i \(0.766822\pi\)
\(504\) 0 0
\(505\) −94170.3 + 55916.8i −0.369259 + 0.219260i
\(506\) 0 0
\(507\) 73343.9 73343.9i 0.285330 0.285330i
\(508\) 0 0
\(509\) 294705.i 1.13750i 0.822510 + 0.568750i \(0.192573\pi\)
−0.822510 + 0.568750i \(0.807427\pi\)
\(510\) 0 0
\(511\) 24094.7 0.0922739
\(512\) 0 0
\(513\) −25812.3 25812.3i −0.0980827 0.0980827i
\(514\) 0 0
\(515\) 234509. + 59770.5i 0.884187 + 0.225358i
\(516\) 0 0
\(517\) −84486.0 + 84486.0i −0.316085 + 0.316085i
\(518\) 0 0
\(519\) 80748.6i 0.299778i
\(520\) 0 0
\(521\) 284288. 1.04733 0.523664 0.851925i \(-0.324565\pi\)
0.523664 + 0.851925i \(0.324565\pi\)
\(522\) 0 0
\(523\) −121118. 121118.i −0.442796 0.442796i 0.450154 0.892951i \(-0.351369\pi\)
−0.892951 + 0.450154i \(0.851369\pi\)
\(524\) 0 0
\(525\) −64278.2 + 117906.i −0.233209 + 0.427776i
\(526\) 0 0
\(527\) −258402. + 258402.i −0.930411 + 0.930411i
\(528\) 0 0
\(529\) 191585.i 0.684621i
\(530\) 0 0
\(531\) −133209. −0.472439
\(532\) 0 0
\(533\) −205206. 205206.i −0.722331 0.722331i
\(534\) 0 0
\(535\) −111014. + 435563.i −0.387857 + 1.52175i
\(536\) 0 0
\(537\) 205373. 205373.i 0.712188 0.712188i
\(538\) 0 0
\(539\) 69673.0i 0.239821i
\(540\) 0 0
\(541\) −434885. −1.48587 −0.742933 0.669366i \(-0.766566\pi\)
−0.742933 + 0.669366i \(0.766566\pi\)
\(542\) 0 0
\(543\) 225858. + 225858.i 0.766014 + 0.766014i
\(544\) 0 0
\(545\) 151196. + 254631.i 0.509034 + 0.857272i
\(546\) 0 0
\(547\) 7758.44 7758.44i 0.0259298 0.0259298i −0.694023 0.719953i \(-0.744163\pi\)
0.719953 + 0.694023i \(0.244163\pi\)
\(548\) 0 0
\(549\) 158574.i 0.526124i
\(550\) 0 0
\(551\) 130919. 0.431222
\(552\) 0 0
\(553\) −253178. 253178.i −0.827895 0.827895i
\(554\) 0 0
\(555\) −286240. + 169964.i −0.929274 + 0.551788i
\(556\) 0 0
\(557\) −22166.3 + 22166.3i −0.0714470 + 0.0714470i −0.741927 0.670480i \(-0.766088\pi\)
0.670480 + 0.741927i \(0.266088\pi\)
\(558\) 0 0
\(559\) 280661.i 0.898168i
\(560\) 0 0
\(561\) 152848. 0.485662
\(562\) 0 0
\(563\) 253305. + 253305.i 0.799147 + 0.799147i 0.982961 0.183814i \(-0.0588445\pi\)
−0.183814 + 0.982961i \(0.558844\pi\)
\(564\) 0 0
\(565\) −575528. 146688.i −1.80289 0.459513i
\(566\) 0 0
\(567\) 21315.3 21315.3i 0.0663017 0.0663017i
\(568\) 0 0
\(569\) 4947.07i 0.0152800i −0.999971 0.00764001i \(-0.997568\pi\)
0.999971 0.00764001i \(-0.00243191\pi\)
\(570\) 0 0
\(571\) −205345. −0.629813 −0.314907 0.949123i \(-0.601973\pi\)
−0.314907 + 0.949123i \(0.601973\pi\)
\(572\) 0 0
\(573\) 77396.1 + 77396.1i 0.235727 + 0.235727i
\(574\) 0 0
\(575\) 178118. 52430.7i 0.538731 0.158581i
\(576\) 0 0
\(577\) −178874. + 178874.i −0.537274 + 0.537274i −0.922727 0.385453i \(-0.874045\pi\)
0.385453 + 0.922727i \(0.374045\pi\)
\(578\) 0 0
\(579\) 120637.i 0.359851i
\(580\) 0 0
\(581\) −305437. −0.904835
\(582\) 0 0
\(583\) −89063.4 89063.4i −0.262037 0.262037i
\(584\) 0 0
\(585\) 15459.5 60655.3i 0.0451736 0.177238i
\(586\) 0 0
\(587\) −240208. + 240208.i −0.697126 + 0.697126i −0.963790 0.266663i \(-0.914079\pi\)
0.266663 + 0.963790i \(0.414079\pi\)
\(588\) 0 0
\(589\) 325849.i 0.939260i
\(590\) 0 0
\(591\) 272512. 0.780208
\(592\) 0 0
\(593\) −481681. 481681.i −1.36978 1.36978i −0.860752 0.509025i \(-0.830006\pi\)
−0.509025 0.860752i \(-0.669994\pi\)
\(594\) 0 0
\(595\) 154012. + 259375.i 0.435033 + 0.732645i
\(596\) 0 0
\(597\) −57236.1 + 57236.1i −0.160591 + 0.160591i
\(598\) 0 0
\(599\) 154606.i 0.430896i −0.976515 0.215448i \(-0.930879\pi\)
0.976515 0.215448i \(-0.0691211\pi\)
\(600\) 0 0
\(601\) 439313. 1.21626 0.608128 0.793839i \(-0.291921\pi\)
0.608128 + 0.793839i \(0.291921\pi\)
\(602\) 0 0
\(603\) −25179.1 25179.1i −0.0692478 0.0692478i
\(604\) 0 0
\(605\) 96283.2 57171.4i 0.263051 0.156195i
\(606\) 0 0
\(607\) −37796.8 + 37796.8i −0.102584 + 0.102584i −0.756536 0.653952i \(-0.773110\pi\)
0.653952 + 0.756536i \(0.273110\pi\)
\(608\) 0 0
\(609\) 108110.i 0.291496i
\(610\) 0 0
\(611\) 109912. 0.294417
\(612\) 0 0
\(613\) 140335. + 140335.i 0.373461 + 0.373461i 0.868736 0.495275i \(-0.164933\pi\)
−0.495275 + 0.868736i \(0.664933\pi\)
\(614\) 0 0
\(615\) 393939. + 100405.i 1.04155 + 0.265464i
\(616\) 0 0
\(617\) 54366.7 54366.7i 0.142811 0.142811i −0.632086 0.774898i \(-0.717801\pi\)
0.774898 + 0.632086i \(0.217801\pi\)
\(618\) 0 0
\(619\) 273964.i 0.715011i −0.933911 0.357505i \(-0.883627\pi\)
0.933911 0.357505i \(-0.116373\pi\)
\(620\) 0 0
\(621\) −41679.0 −0.108077
\(622\) 0 0
\(623\) 149183. + 149183.i 0.384365 + 0.384365i
\(624\) 0 0
\(625\) −211631. 328329.i −0.541776 0.840523i
\(626\) 0 0
\(627\) 96371.9 96371.9i 0.245141 0.245141i
\(628\) 0 0
\(629\) 747790.i 1.89007i
\(630\) 0 0
\(631\) −770840. −1.93600 −0.968001 0.250947i \(-0.919258\pi\)
−0.968001 + 0.250947i \(0.919258\pi\)
\(632\) 0 0
\(633\) 263334. + 263334.i 0.657204 + 0.657204i
\(634\) 0 0
\(635\) 130541. 512177.i 0.323743 1.27020i
\(636\) 0 0
\(637\) −45320.5 + 45320.5i −0.111690 + 0.111690i
\(638\) 0 0
\(639\) 119879.i 0.293590i
\(640\) 0 0
\(641\) 404369. 0.984152 0.492076 0.870552i \(-0.336238\pi\)
0.492076 + 0.870552i \(0.336238\pi\)
\(642\) 0 0
\(643\) 438578. + 438578.i 1.06078 + 1.06078i 0.998029 + 0.0627491i \(0.0199868\pi\)
0.0627491 + 0.998029i \(0.480013\pi\)
\(644\) 0 0
\(645\) −200733. 338057.i −0.482502 0.812588i
\(646\) 0 0
\(647\) 190458. 190458.i 0.454978 0.454978i −0.442025 0.897003i \(-0.645740\pi\)
0.897003 + 0.442025i \(0.145740\pi\)
\(648\) 0 0
\(649\) 497346.i 1.18078i
\(650\) 0 0
\(651\) −269079. −0.634919
\(652\) 0 0
\(653\) −184154. 184154.i −0.431872 0.431872i 0.457393 0.889265i \(-0.348783\pi\)
−0.889265 + 0.457393i \(0.848783\pi\)
\(654\) 0 0
\(655\) 80858.8 48012.6i 0.188471 0.111911i
\(656\) 0 0
\(657\) 11124.8 11124.8i 0.0257728 0.0257728i
\(658\) 0 0
\(659\) 136158.i 0.313525i −0.987636 0.156762i \(-0.949894\pi\)
0.987636 0.156762i \(-0.0501057\pi\)
\(660\) 0 0
\(661\) −302378. −0.692065 −0.346032 0.938223i \(-0.612471\pi\)
−0.346032 + 0.938223i \(0.612471\pi\)
\(662\) 0 0
\(663\) −99423.6 99423.6i −0.226184 0.226184i
\(664\) 0 0
\(665\) 260644. + 66431.7i 0.589392 + 0.150222i
\(666\) 0 0
\(667\) 105697. 105697.i 0.237582 0.237582i
\(668\) 0 0
\(669\) 79614.5i 0.177885i
\(670\) 0 0
\(671\) 592047. 1.31496
\(672\) 0 0
\(673\) −273891. 273891.i −0.604710 0.604710i 0.336849 0.941559i \(-0.390639\pi\)
−0.941559 + 0.336849i \(0.890639\pi\)
\(674\) 0 0
\(675\) 24760.5 + 84116.5i 0.0543441 + 0.184618i
\(676\) 0 0
\(677\) 178343. 178343.i 0.389116 0.389116i −0.485256 0.874372i \(-0.661274\pi\)
0.874372 + 0.485256i \(0.161274\pi\)
\(678\) 0 0
\(679\) 141445.i 0.306795i
\(680\) 0 0
\(681\) 62727.3 0.135258
\(682\) 0 0
\(683\) 238606. + 238606.i 0.511494 + 0.511494i 0.914984 0.403490i \(-0.132203\pi\)
−0.403490 + 0.914984i \(0.632203\pi\)
\(684\) 0 0
\(685\) 120319. 472071.i 0.256421 1.00607i
\(686\) 0 0
\(687\) 195367. 195367.i 0.413941 0.413941i
\(688\) 0 0
\(689\) 115867.i 0.244074i
\(690\) 0 0
\(691\) 207226. 0.433998 0.216999 0.976172i \(-0.430373\pi\)
0.216999 + 0.976172i \(0.430373\pi\)
\(692\) 0 0
\(693\) 79581.8 + 79581.8i 0.165710 + 0.165710i
\(694\) 0 0
\(695\) −322720. 543497.i −0.668123 1.12520i
\(696\) 0 0
\(697\) 645727. 645727.i 1.32918 1.32918i
\(698\) 0 0
\(699\) 60820.5i 0.124479i
\(700\) 0 0
\(701\) −472139. −0.960803 −0.480401 0.877049i \(-0.659509\pi\)
−0.480401 + 0.877049i \(0.659509\pi\)
\(702\) 0 0
\(703\) 471488. + 471488.i 0.954025 + 0.954025i
\(704\) 0 0
\(705\) −132389. + 78610.7i −0.266364 + 0.158162i
\(706\) 0 0
\(707\) −128091. + 128091.i −0.256259 + 0.256259i
\(708\) 0 0
\(709\) 180630.i 0.359333i 0.983728 + 0.179667i \(0.0575019\pi\)
−0.983728 + 0.179667i \(0.942498\pi\)
\(710\) 0 0
\(711\) −233790. −0.462474
\(712\) 0 0
\(713\) 263073. + 263073.i 0.517485 + 0.517485i
\(714\) 0 0
\(715\) 226460. + 57719.2i 0.442976 + 0.112904i
\(716\) 0 0
\(717\) −235911. + 235911.i −0.458892 + 0.458892i
\(718\) 0 0
\(719\) 71380.5i 0.138077i 0.997614 + 0.0690386i \(0.0219932\pi\)
−0.997614 + 0.0690386i \(0.978007\pi\)
\(720\) 0 0
\(721\) 400280. 0.770004
\(722\) 0 0
\(723\) −193971. 193971.i −0.371073 0.371073i
\(724\) 0 0
\(725\) −276111. 150526.i −0.525299 0.286375i
\(726\) 0 0
\(727\) −520312. + 520312.i −0.984453 + 0.984453i −0.999881 0.0154283i \(-0.995089\pi\)
0.0154283 + 0.999881i \(0.495089\pi\)
\(728\) 0 0
\(729\) 19683.0i 0.0370370i
\(730\) 0 0
\(731\) −883162. −1.65274
\(732\) 0 0
\(733\) 304781. + 304781.i 0.567258 + 0.567258i 0.931359 0.364101i \(-0.118624\pi\)
−0.364101 + 0.931359i \(0.618624\pi\)
\(734\) 0 0
\(735\) 22174.8 87002.6i 0.0410474 0.161049i
\(736\) 0 0
\(737\) 94007.8 94007.8i 0.173073 0.173073i
\(738\) 0 0
\(739\) 422739.i 0.774076i 0.922064 + 0.387038i \(0.126502\pi\)
−0.922064 + 0.387038i \(0.873498\pi\)
\(740\) 0 0
\(741\) −125375. −0.228336
\(742\) 0 0
\(743\) −309220. 309220.i −0.560131 0.560131i 0.369214 0.929344i \(-0.379627\pi\)
−0.929344 + 0.369214i \(0.879627\pi\)
\(744\) 0 0
\(745\) 173508. + 292208.i 0.312613 + 0.526477i
\(746\) 0 0
\(747\) −141024. + 141024.i −0.252727 + 0.252727i
\(748\) 0 0
\(749\) 743457.i 1.32523i
\(750\) 0 0
\(751\) −830203. −1.47199 −0.735994 0.676988i \(-0.763285\pi\)
−0.735994 + 0.676988i \(0.763285\pi\)
\(752\) 0 0
\(753\) 27995.9 + 27995.9i 0.0493747 + 0.0493747i
\(754\) 0 0
\(755\) −379578. + 225387.i −0.665896 + 0.395398i
\(756\) 0 0
\(757\) 319294. 319294.i 0.557184 0.557184i −0.371321 0.928505i \(-0.621095\pi\)
0.928505 + 0.371321i \(0.121095\pi\)
\(758\) 0 0
\(759\) 155611.i 0.270120i
\(760\) 0 0
\(761\) −668943. −1.15510 −0.577550 0.816355i \(-0.695991\pi\)
−0.577550 + 0.816355i \(0.695991\pi\)
\(762\) 0 0
\(763\) 346351. + 346351.i 0.594931 + 0.594931i
\(764\) 0 0
\(765\) 190866. + 48646.9i 0.326140 + 0.0831252i
\(766\) 0 0
\(767\) −323510. + 323510.i −0.549917 + 0.549917i
\(768\) 0 0
\(769\) 831330.i 1.40579i 0.711293 + 0.702896i \(0.248110\pi\)
−0.711293 + 0.702896i \(0.751890\pi\)
\(770\) 0 0
\(771\) −418129. −0.703398
\(772\) 0 0
\(773\) 457925. + 457925.i 0.766364 + 0.766364i 0.977464 0.211100i \(-0.0677047\pi\)
−0.211100 + 0.977464i \(0.567705\pi\)
\(774\) 0 0
\(775\) 374649. 687220.i 0.623765 1.14417i
\(776\) 0 0
\(777\) −389344. + 389344.i −0.644899 + 0.644899i
\(778\) 0 0
\(779\) 814273.i 1.34182i
\(780\) 0 0
\(781\) 447575. 0.733777
\(782\) 0 0
\(783\) 49915.8 + 49915.8i 0.0814169 + 0.0814169i
\(784\) 0 0
\(785\) −26079.8 + 102324.i −0.0423218 + 0.166049i
\(786\) 0 0
\(787\) 440089. 440089.i 0.710544 0.710544i −0.256105 0.966649i \(-0.582439\pi\)
0.966649 + 0.256105i \(0.0824394\pi\)
\(788\) 0 0
\(789\) 577333.i 0.927411i
\(790\) 0 0
\(791\) −982362. −1.57007
\(792\) 0 0
\(793\) −385111. 385111.i −0.612406 0.612406i
\(794\) 0 0
\(795\) −82869.8 139562.i −0.131118 0.220818i
\(796\) 0 0
\(797\) 874514. 874514.i 1.37673 1.37673i 0.526654 0.850080i \(-0.323446\pi\)
0.850080 0.526654i \(-0.176554\pi\)
\(798\) 0 0
\(799\) 345862.i 0.541764i
\(800\) 0 0
\(801\) 137759. 0.214712
\(802\) 0 0
\(803\) 41535.1 + 41535.1i 0.0644146 + 0.0644146i
\(804\) 0 0
\(805\) 264064. 156797.i 0.407490 0.241961i
\(806\) 0 0
\(807\) 399893. 399893.i 0.614041 0.614041i
\(808\) 0 0
\(809\) 900682.i 1.37618i 0.725627 + 0.688088i \(0.241550\pi\)
−0.725627 + 0.688088i \(0.758450\pi\)
\(810\) 0 0
\(811\) 560145. 0.851645 0.425822 0.904807i \(-0.359985\pi\)
0.425822 + 0.904807i \(0.359985\pi\)
\(812\) 0 0
\(813\) −187685. 187685.i −0.283954 0.283954i
\(814\) 0 0
\(815\) 499918. + 127417.i 0.752634 + 0.191828i
\(816\) 0 0
\(817\) −556841. + 556841.i −0.834232 + 0.834232i
\(818\) 0 0
\(819\) 103532.i 0.154350i
\(820\) 0 0
\(821\) −391487. −0.580805 −0.290403 0.956905i \(-0.593789\pi\)
−0.290403 + 0.956905i \(0.593789\pi\)
\(822\) 0 0
\(823\) −877887. 877887.i −1.29610 1.29610i −0.930947 0.365154i \(-0.881017\pi\)
−0.365154 0.930947i \(-0.618983\pi\)
\(824\) 0 0
\(825\) −314054. + 92444.9i −0.461420 + 0.135824i
\(826\) 0 0
\(827\) −796044. + 796044.i −1.16393 + 1.16393i −0.180320 + 0.983608i \(0.557713\pi\)
−0.983608 + 0.180320i \(0.942287\pi\)
\(828\) 0 0
\(829\) 83808.3i 0.121949i −0.998139 0.0609745i \(-0.980579\pi\)
0.998139 0.0609745i \(-0.0194208\pi\)
\(830\) 0 0
\(831\) −275614. −0.399116
\(832\) 0 0
\(833\) −142611. 142611.i −0.205524 0.205524i
\(834\) 0 0
\(835\) −35576.2 + 139583.i −0.0510254 + 0.200198i
\(836\) 0 0
\(837\) −124237. + 124237.i −0.177337 + 0.177337i
\(838\) 0 0
\(839\) 1.03825e6i 1.47495i −0.675372 0.737477i \(-0.736017\pi\)
0.675372 0.737477i \(-0.263983\pi\)
\(840\) 0 0
\(841\) 454110. 0.642050
\(842\) 0 0
\(843\) −371795. 371795.i −0.523176 0.523176i
\(844\) 0 0
\(845\) 254791. + 429097.i 0.356838 + 0.600956i
\(846\) 0 0
\(847\) 130965. 130965.i 0.182553 0.182553i
\(848\) 0 0
\(849\) 29682.2i 0.0411795i
\(850\) 0 0
\(851\) 761309. 1.05124
\(852\) 0 0
\(853\) 417742. + 417742.i 0.574130 + 0.574130i 0.933280 0.359150i \(-0.116933\pi\)
−0.359150 + 0.933280i \(0.616933\pi\)
\(854\) 0 0
\(855\) 151015. 89670.0i 0.206579 0.122663i
\(856\) 0 0
\(857\) 295564. 295564.i 0.402430 0.402430i −0.476659 0.879088i \(-0.658152\pi\)
0.879088 + 0.476659i \(0.158152\pi\)
\(858\) 0 0
\(859\) 156574.i 0.212194i −0.994356 0.106097i \(-0.966165\pi\)
0.994356 0.106097i \(-0.0338354\pi\)
\(860\) 0 0
\(861\) 672409. 0.907041
\(862\) 0 0
\(863\) 852253. + 852253.i 1.14432 + 1.14432i 0.987652 + 0.156667i \(0.0500748\pi\)
0.156667 + 0.987652i \(0.449925\pi\)
\(864\) 0 0
\(865\) −376466. 95952.0i −0.503146 0.128239i
\(866\) 0 0
\(867\) 5983.02 5983.02i 0.00795944 0.00795944i
\(868\) 0 0
\(869\) 872871.i 1.15587i
\(870\) 0 0
\(871\) −122299. −0.161208
\(872\) 0 0
\(873\) −65306.8 65306.8i −0.0856900 0.0856900i
\(874\) 0 0
\(875\) −473321. 439783.i −0.618215 0.574411i
\(876\) 0 0
\(877\) 673537. 673537.i 0.875713 0.875713i −0.117374 0.993088i \(-0.537448\pi\)
0.993088 + 0.117374i \(0.0374477\pi\)
\(878\) 0 0
\(879\) 680203.i 0.880361i
\(880\) 0 0
\(881\) 801878. 1.03313 0.516567 0.856247i \(-0.327210\pi\)
0.516567 + 0.856247i \(0.327210\pi\)
\(882\) 0 0
\(883\) 419877. + 419877.i 0.538519 + 0.538519i 0.923094 0.384575i \(-0.125652\pi\)
−0.384575 + 0.923094i \(0.625652\pi\)
\(884\) 0 0
\(885\) 158290. 621050.i 0.202101 0.792939i
\(886\) 0 0
\(887\) 700728. 700728.i 0.890640 0.890640i −0.103943 0.994583i \(-0.533146\pi\)
0.994583 + 0.103943i \(0.0331461\pi\)
\(888\) 0 0
\(889\) 874228.i 1.10617i
\(890\) 0 0
\(891\) 73487.7 0.0925677
\(892\) 0 0
\(893\) 218069. + 218069.i 0.273458 + 0.273458i
\(894\) 0 0
\(895\) 713449. + 1.20153e6i 0.890670 + 1.49999i
\(896\) 0 0
\(897\) −101221. + 101221.i −0.125802 + 0.125802i
\(898\) 0 0
\(899\) 630126.i 0.779665i
\(900\) 0 0
\(901\) −364601. −0.449126
\(902\) 0 0
\(903\) −459827. 459827.i −0.563922 0.563922i
\(904\) 0 0
\(905\) −1.32138e6 + 784614.i −1.61336 + 0.957986i
\(906\) 0 0
\(907\) −226251. + 226251.i −0.275027 + 0.275027i −0.831120 0.556093i \(-0.812300\pi\)
0.556093 + 0.831120i \(0.312300\pi\)
\(908\) 0 0
\(909\) 118282.i 0.143150i
\(910\) 0 0
\(911\) 840996. 1.01334 0.506672 0.862139i \(-0.330875\pi\)
0.506672 + 0.862139i \(0.330875\pi\)
\(912\) 0 0
\(913\) −526521. 526521.i −0.631647 0.631647i
\(914\) 0 0
\(915\) 739306. + 188431.i 0.883043 + 0.225066i
\(916\) 0 0
\(917\) 109985. 109985.i 0.130796 0.130796i
\(918\) 0 0
\(919\) 131956.i 0.156242i 0.996944 + 0.0781208i \(0.0248920\pi\)
−0.996944 + 0.0781208i \(0.975108\pi\)
\(920\) 0 0
\(921\) 604900. 0.713123
\(922\) 0 0
\(923\) −291136. 291136.i −0.341737 0.341737i
\(924\) 0 0
\(925\) −452275. 1.53647e6i −0.528591 1.79573i
\(926\) 0 0
\(927\) 184814. 184814.i 0.215068 0.215068i
\(928\) 0 0
\(929\) 789125.i 0.914354i −0.889376 0.457177i \(-0.848861\pi\)
0.889376 0.457177i \(-0.151139\pi\)
\(930\) 0 0
\(931\) −179835. −0.207479
\(932\) 0 0
\(933\) 245788. + 245788.i 0.282356 + 0.282356i
\(934\) 0 0
\(935\) −181626. + 712609.i −0.207757 + 0.815132i
\(936\) 0 0
\(937\) −979642. + 979642.i −1.11580 + 1.11580i −0.123454 + 0.992350i \(0.539397\pi\)
−0.992350 + 0.123454i \(0.960603\pi\)
\(938\) 0 0
\(939\) 94815.1i 0.107534i
\(940\) 0 0
\(941\) −961058. −1.08535 −0.542675 0.839942i \(-0.682589\pi\)
−0.542675 + 0.839942i \(0.682589\pi\)
\(942\) 0 0
\(943\) −657401. 657401.i −0.739277 0.739277i
\(944\) 0 0
\(945\) 74047.5 + 124705.i 0.0829176 + 0.139643i
\(946\) 0 0
\(947\) −96791.3 + 96791.3i −0.107929 + 0.107929i −0.759009 0.651080i \(-0.774316\pi\)
0.651080 + 0.759009i \(0.274316\pi\)
\(948\) 0 0
\(949\) 54035.0i 0.0599988i
\(950\) 0 0
\(951\) 641242. 0.709024
\(952\) 0 0
\(953\) 280586. + 280586.i 0.308944 + 0.308944i 0.844500 0.535556i \(-0.179898\pi\)
−0.535556 + 0.844500i \(0.679898\pi\)
\(954\) 0 0
\(955\) −452804. + 268868.i −0.496482 + 0.294803i
\(956\) 0 0
\(957\) −186364. + 186364.i −0.203487 + 0.203487i
\(958\) 0 0
\(959\) 805772.i 0.876143i
\(960\) 0 0
\(961\) 644819. 0.698218
\(962\) 0 0
\(963\) 343263. + 343263.i 0.370147 + 0.370147i
\(964\) 0 0
\(965\) 562433. + 143350.i 0.603972 + 0.153938i
\(966\) 0 0
\(967\) −1.05482e6 + 1.05482e6i −1.12805 + 1.12805i −0.137551 + 0.990495i \(0.543923\pi\)
−0.990495 + 0.137551i \(0.956077\pi\)
\(968\) 0 0
\(969\) 394520.i 0.420167i
\(970\) 0 0
\(971\) 494524. 0.524504 0.262252 0.964999i \(-0.415535\pi\)
0.262252 + 0.964999i \(0.415535\pi\)
\(972\) 0 0
\(973\) −739268. 739268.i −0.780865 0.780865i
\(974\) 0 0
\(975\) 264417. + 144151.i 0.278151 + 0.151638i
\(976\) 0 0
\(977\) −391473. + 391473.i −0.410121 + 0.410121i −0.881781 0.471660i \(-0.843655\pi\)
0.471660 + 0.881781i \(0.343655\pi\)
\(978\) 0 0
\(979\) 514333.i 0.536635i
\(980\) 0 0
\(981\) 319828. 0.332337
\(982\) 0 0
\(983\) 657131. + 657131.i 0.680057 + 0.680057i 0.960013 0.279956i \(-0.0903200\pi\)
−0.279956 + 0.960013i \(0.590320\pi\)
\(984\) 0 0
\(985\) −323821. + 1.27051e6i −0.333758 + 1.30950i
\(986\) 0 0
\(987\) −180077. + 180077.i −0.184852 + 0.184852i
\(988\) 0 0
\(989\) 899128.i 0.919240i
\(990\) 0 0
\(991\) 443257. 0.451344 0.225672 0.974203i \(-0.427542\pi\)
0.225672 + 0.974203i \(0.427542\pi\)
\(992\) 0 0
\(993\) −328980. 328980.i −0.333634 0.333634i
\(994\) 0 0
\(995\) −198834. 334859.i −0.200837 0.338233i
\(996\) 0 0
\(997\) −667384. + 667384.i −0.671407 + 0.671407i −0.958040 0.286634i \(-0.907464\pi\)
0.286634 + 0.958040i \(0.407464\pi\)
\(998\) 0 0
\(999\) 359530.i 0.360250i
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 60.5.k.a.13.3 8
3.2 odd 2 180.5.l.b.73.4 8
4.3 odd 2 240.5.bg.d.193.1 8
5.2 odd 4 inner 60.5.k.a.37.3 yes 8
5.3 odd 4 300.5.k.d.157.2 8
5.4 even 2 300.5.k.d.193.2 8
15.2 even 4 180.5.l.b.37.4 8
15.8 even 4 900.5.l.k.757.3 8
15.14 odd 2 900.5.l.k.793.3 8
20.7 even 4 240.5.bg.d.97.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.5.k.a.13.3 8 1.1 even 1 trivial
60.5.k.a.37.3 yes 8 5.2 odd 4 inner
180.5.l.b.37.4 8 15.2 even 4
180.5.l.b.73.4 8 3.2 odd 2
240.5.bg.d.97.1 8 20.7 even 4
240.5.bg.d.193.1 8 4.3 odd 2
300.5.k.d.157.2 8 5.3 odd 4
300.5.k.d.193.2 8 5.4 even 2
900.5.l.k.757.3 8 15.8 even 4
900.5.l.k.793.3 8 15.14 odd 2