Properties

Label 117.5.j.b.109.9
Level $117$
Weight $5$
Character 117.109
Analytic conductor $12.094$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,5,Mod(73,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.73");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 117.j (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: \(12.0942856808\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 5446 x^{16} - 1452 x^{15} + 106320 x^{13} + 8376897 x^{12} - 1643220 x^{11} + 1054152 x^{10} + \cdots + 2103506496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{10} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 109.9
Root \(4.32919 - 4.32919i\) of defining polynomial
Character \(\chi\) \(=\) 117.109
Dual form 117.5.j.b.73.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(4.32919 - 4.32919i) q^{2} -21.4837i q^{4} +(26.0804 - 26.0804i) q^{5} +(-16.9579 - 16.9579i) q^{7} +(-23.7399 - 23.7399i) q^{8} -225.814i q^{10} +(69.5293 + 69.5293i) q^{11} +(-93.5895 - 140.720i) q^{13} -146.828 q^{14} +138.190 q^{16} -166.635i q^{17} +(-468.820 + 468.820i) q^{19} +(-560.303 - 560.303i) q^{20} +602.011 q^{22} -388.867i q^{23} -735.373i q^{25} +(-1014.37 - 204.035i) q^{26} +(-364.318 + 364.318i) q^{28} +767.891 q^{29} +(-66.8738 + 66.8738i) q^{31} +(978.089 - 978.089i) q^{32} +(-721.395 - 721.395i) q^{34} -884.537 q^{35} +(1532.33 + 1532.33i) q^{37} +4059.22i q^{38} -1238.29 q^{40} +(-802.986 + 802.986i) q^{41} +2798.62i q^{43} +(1493.75 - 1493.75i) q^{44} +(-1683.48 - 1683.48i) q^{46} +(-33.8918 - 33.8918i) q^{47} -1825.86i q^{49} +(-3183.56 - 3183.56i) q^{50} +(-3023.18 + 2010.65i) q^{52} +3139.54 q^{53} +3626.70 q^{55} +805.159i q^{56} +(3324.34 - 3324.34i) q^{58} +(3959.16 + 3959.16i) q^{59} +504.230 q^{61} +579.018i q^{62} -6257.62i q^{64} +(-6110.87 - 1229.17i) q^{65} +(4218.53 - 4218.53i) q^{67} -3579.94 q^{68} +(-3829.33 + 3829.33i) q^{70} +(3025.86 - 3025.86i) q^{71} +(3088.98 + 3088.98i) q^{73} +13267.5 q^{74} +(10072.0 + 10072.0i) q^{76} -2358.14i q^{77} -9376.55 q^{79} +(3604.05 - 3604.05i) q^{80} +6952.55i q^{82} +(-6229.50 + 6229.50i) q^{83} +(-4345.91 - 4345.91i) q^{85} +(12115.7 + 12115.7i) q^{86} -3301.24i q^{88} +(-8558.42 - 8558.42i) q^{89} +(-799.227 + 3973.39i) q^{91} -8354.30 q^{92} -293.448 q^{94} +24454.0i q^{95} +(2882.26 - 2882.26i) q^{97} +(-7904.48 - 7904.48i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 24 q^{5} - 24 q^{7} - 372 q^{11} - 224 q^{13} - 480 q^{14} - 2328 q^{16} - 840 q^{19} - 228 q^{20} + 3536 q^{22} + 828 q^{26} - 1984 q^{28} + 5064 q^{29} + 1712 q^{31} + 7260 q^{32} + 8040 q^{34}+ \cdots - 11544 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.32919 4.32919i 1.08230 1.08230i 0.0860014 0.996295i \(-0.472591\pi\)
0.996295 0.0860014i \(-0.0274090\pi\)
\(3\) 0 0
\(4\) 21.4837i 1.34273i
\(5\) 26.0804 26.0804i 1.04322 1.04322i 0.0441922 0.999023i \(-0.485929\pi\)
0.999023 0.0441922i \(-0.0140714\pi\)
\(6\) 0 0
\(7\) −16.9579 16.9579i −0.346080 0.346080i 0.512567 0.858647i \(-0.328694\pi\)
−0.858647 + 0.512567i \(0.828694\pi\)
\(8\) −23.7399 23.7399i −0.370937 0.370937i
\(9\) 0 0
\(10\) 225.814i 2.25814i
\(11\) 69.5293 + 69.5293i 0.574623 + 0.574623i 0.933417 0.358794i \(-0.116812\pi\)
−0.358794 + 0.933417i \(0.616812\pi\)
\(12\) 0 0
\(13\) −93.5895 140.720i −0.553784 0.832660i
\(14\) −146.828 −0.749121
\(15\) 0 0
\(16\) 138.190 0.539804
\(17\) 166.635i 0.576593i −0.957541 0.288296i \(-0.906911\pi\)
0.957541 0.288296i \(-0.0930888\pi\)
\(18\) 0 0
\(19\) −468.820 + 468.820i −1.29867 + 1.29867i −0.369401 + 0.929270i \(0.620437\pi\)
−0.929270 + 0.369401i \(0.879563\pi\)
\(20\) −560.303 560.303i −1.40076 1.40076i
\(21\) 0 0
\(22\) 602.011 1.24382
\(23\) 388.867i 0.735098i −0.930004 0.367549i \(-0.880197\pi\)
0.930004 0.367549i \(-0.119803\pi\)
\(24\) 0 0
\(25\) 735.373i 1.17660i
\(26\) −1014.37 204.035i −1.50054 0.301827i
\(27\) 0 0
\(28\) −364.318 + 364.318i −0.464692 + 0.464692i
\(29\) 767.891 0.913069 0.456534 0.889706i \(-0.349091\pi\)
0.456534 + 0.889706i \(0.349091\pi\)
\(30\) 0 0
\(31\) −66.8738 + 66.8738i −0.0695877 + 0.0695877i −0.741044 0.671456i \(-0.765669\pi\)
0.671456 + 0.741044i \(0.265669\pi\)
\(32\) 978.089 978.089i 0.955165 0.955165i
\(33\) 0 0
\(34\) −721.395 721.395i −0.624044 0.624044i
\(35\) −884.537 −0.722071
\(36\) 0 0
\(37\) 1532.33 + 1532.33i 1.11931 + 1.11931i 0.991843 + 0.127465i \(0.0406841\pi\)
0.127465 + 0.991843i \(0.459316\pi\)
\(38\) 4059.22i 2.81109i
\(39\) 0 0
\(40\) −1238.29 −0.773933
\(41\) −802.986 + 802.986i −0.477684 + 0.477684i −0.904390 0.426707i \(-0.859674\pi\)
0.426707 + 0.904390i \(0.359674\pi\)
\(42\) 0 0
\(43\) 2798.62i 1.51358i 0.653655 + 0.756792i \(0.273235\pi\)
−0.653655 + 0.756792i \(0.726765\pi\)
\(44\) 1493.75 1493.75i 0.771564 0.771564i
\(45\) 0 0
\(46\) −1683.48 1683.48i −0.795594 0.795594i
\(47\) −33.8918 33.8918i −0.0153426 0.0153426i 0.699394 0.714736i \(-0.253453\pi\)
−0.714736 + 0.699394i \(0.753453\pi\)
\(48\) 0 0
\(49\) 1825.86i 0.760458i
\(50\) −3183.56 3183.56i −1.27343 1.27343i
\(51\) 0 0
\(52\) −3023.18 + 2010.65i −1.11804 + 0.743583i
\(53\) 3139.54 1.11767 0.558836 0.829278i \(-0.311248\pi\)
0.558836 + 0.829278i \(0.311248\pi\)
\(54\) 0 0
\(55\) 3626.70 1.19891
\(56\) 805.159i 0.256747i
\(57\) 0 0
\(58\) 3324.34 3324.34i 0.988211 0.988211i
\(59\) 3959.16 + 3959.16i 1.13736 + 1.13736i 0.988921 + 0.148440i \(0.0474253\pi\)
0.148440 + 0.988921i \(0.452575\pi\)
\(60\) 0 0
\(61\) 504.230 0.135509 0.0677546 0.997702i \(-0.478417\pi\)
0.0677546 + 0.997702i \(0.478417\pi\)
\(62\) 579.018i 0.150629i
\(63\) 0 0
\(64\) 6257.62i 1.52774i
\(65\) −6110.87 1229.17i −1.44636 0.290928i
\(66\) 0 0
\(67\) 4218.53 4218.53i 0.939748 0.939748i −0.0585372 0.998285i \(-0.518644\pi\)
0.998285 + 0.0585372i \(0.0186436\pi\)
\(68\) −3579.94 −0.774209
\(69\) 0 0
\(70\) −3829.33 + 3829.33i −0.781495 + 0.781495i
\(71\) 3025.86 3025.86i 0.600249 0.600249i −0.340130 0.940379i \(-0.610471\pi\)
0.940379 + 0.340130i \(0.110471\pi\)
\(72\) 0 0
\(73\) 3088.98 + 3088.98i 0.579654 + 0.579654i 0.934808 0.355154i \(-0.115571\pi\)
−0.355154 + 0.934808i \(0.615571\pi\)
\(74\) 13267.5 2.42285
\(75\) 0 0
\(76\) 10072.0 + 10072.0i 1.74377 + 1.74377i
\(77\) 2358.14i 0.397730i
\(78\) 0 0
\(79\) −9376.55 −1.50241 −0.751205 0.660069i \(-0.770527\pi\)
−0.751205 + 0.660069i \(0.770527\pi\)
\(80\) 3604.05 3604.05i 0.563132 0.563132i
\(81\) 0 0
\(82\) 6952.55i 1.03399i
\(83\) −6229.50 + 6229.50i −0.904267 + 0.904267i −0.995802 0.0915344i \(-0.970823\pi\)
0.0915344 + 0.995802i \(0.470823\pi\)
\(84\) 0 0
\(85\) −4345.91 4345.91i −0.601510 0.601510i
\(86\) 12115.7 + 12115.7i 1.63815 + 1.63815i
\(87\) 0 0
\(88\) 3301.24i 0.426297i
\(89\) −8558.42 8558.42i −1.08047 1.08047i −0.996465 0.0840075i \(-0.973228\pi\)
−0.0840075 0.996465i \(-0.526772\pi\)
\(90\) 0 0
\(91\) −799.227 + 3973.39i −0.0965133 + 0.479820i
\(92\) −8354.30 −0.987039
\(93\) 0 0
\(94\) −293.448 −0.0332105
\(95\) 24454.0i 2.70959i
\(96\) 0 0
\(97\) 2882.26 2882.26i 0.306330 0.306330i −0.537154 0.843484i \(-0.680501\pi\)
0.843484 + 0.537154i \(0.180501\pi\)
\(98\) −7904.48 7904.48i −0.823041 0.823041i
\(99\) 0 0
\(100\) −15798.5 −1.57985
\(101\) 5594.65i 0.548441i 0.961667 + 0.274220i \(0.0884198\pi\)
−0.961667 + 0.274220i \(0.911580\pi\)
\(102\) 0 0
\(103\) 6792.74i 0.640281i 0.947370 + 0.320140i \(0.103730\pi\)
−0.947370 + 0.320140i \(0.896270\pi\)
\(104\) −1118.86 + 5562.48i −0.103445 + 0.514283i
\(105\) 0 0
\(106\) 13591.7 13591.7i 1.20965 1.20965i
\(107\) 8462.24 0.739125 0.369563 0.929206i \(-0.379508\pi\)
0.369563 + 0.929206i \(0.379508\pi\)
\(108\) 0 0
\(109\) −12077.9 + 12077.9i −1.01657 + 1.01657i −0.0167121 + 0.999860i \(0.505320\pi\)
−0.999860 + 0.0167121i \(0.994680\pi\)
\(110\) 15700.7 15700.7i 1.29758 1.29758i
\(111\) 0 0
\(112\) −2343.41 2343.41i −0.186815 0.186815i
\(113\) −14864.2 −1.16409 −0.582043 0.813158i \(-0.697746\pi\)
−0.582043 + 0.813158i \(0.697746\pi\)
\(114\) 0 0
\(115\) −10141.8 10141.8i −0.766866 0.766866i
\(116\) 16497.1i 1.22601i
\(117\) 0 0
\(118\) 34279.8 2.46192
\(119\) −2825.78 + 2825.78i −0.199547 + 0.199547i
\(120\) 0 0
\(121\) 4972.34i 0.339618i
\(122\) 2182.90 2182.90i 0.146661 0.146661i
\(123\) 0 0
\(124\) 1436.70 + 1436.70i 0.0934376 + 0.0934376i
\(125\) −2878.56 2878.56i −0.184228 0.184228i
\(126\) 0 0
\(127\) 10995.6i 0.681728i 0.940113 + 0.340864i \(0.110720\pi\)
−0.940113 + 0.340864i \(0.889280\pi\)
\(128\) −11441.0 11441.0i −0.698301 0.698301i
\(129\) 0 0
\(130\) −31776.4 + 21133.8i −1.88026 + 1.25052i
\(131\) −14023.3 −0.817160 −0.408580 0.912722i \(-0.633976\pi\)
−0.408580 + 0.912722i \(0.633976\pi\)
\(132\) 0 0
\(133\) 15900.4 0.898887
\(134\) 36525.6i 2.03417i
\(135\) 0 0
\(136\) −3955.91 + 3955.91i −0.213879 + 0.213879i
\(137\) −20682.4 20682.4i −1.10194 1.10194i −0.994176 0.107769i \(-0.965629\pi\)
−0.107769 0.994176i \(-0.534371\pi\)
\(138\) 0 0
\(139\) −12584.1 −0.651317 −0.325658 0.945487i \(-0.605586\pi\)
−0.325658 + 0.945487i \(0.605586\pi\)
\(140\) 19003.1i 0.969547i
\(141\) 0 0
\(142\) 26199.0i 1.29929i
\(143\) 3276.92 16291.4i 0.160248 0.796682i
\(144\) 0 0
\(145\) 20026.9 20026.9i 0.952527 0.952527i
\(146\) 26745.5 1.25472
\(147\) 0 0
\(148\) 32920.2 32920.2i 1.50293 1.50293i
\(149\) 1317.22 1317.22i 0.0593315 0.0593315i −0.676818 0.736150i \(-0.736642\pi\)
0.736150 + 0.676818i \(0.236642\pi\)
\(150\) 0 0
\(151\) −6003.08 6003.08i −0.263282 0.263282i 0.563104 0.826386i \(-0.309607\pi\)
−0.826386 + 0.563104i \(0.809607\pi\)
\(152\) 22259.5 0.963449
\(153\) 0 0
\(154\) −10208.8 10208.8i −0.430462 0.430462i
\(155\) 3488.19i 0.145190i
\(156\) 0 0
\(157\) −24232.6 −0.983107 −0.491553 0.870847i \(-0.663571\pi\)
−0.491553 + 0.870847i \(0.663571\pi\)
\(158\) −40592.8 + 40592.8i −1.62605 + 1.62605i
\(159\) 0 0
\(160\) 51017.9i 1.99289i
\(161\) −6594.37 + 6594.37i −0.254403 + 0.254403i
\(162\) 0 0
\(163\) −24239.5 24239.5i −0.912322 0.912322i 0.0841325 0.996455i \(-0.473188\pi\)
−0.996455 + 0.0841325i \(0.973188\pi\)
\(164\) 17251.1 + 17251.1i 0.641401 + 0.641401i
\(165\) 0 0
\(166\) 53937.3i 1.95737i
\(167\) 17352.4 + 17352.4i 0.622195 + 0.622195i 0.946092 0.323897i \(-0.104993\pi\)
−0.323897 + 0.946092i \(0.604993\pi\)
\(168\) 0 0
\(169\) −11043.0 + 26339.8i −0.386646 + 0.922228i
\(170\) −37628.5 −1.30202
\(171\) 0 0
\(172\) 60124.7 2.03234
\(173\) 2780.29i 0.0928961i −0.998921 0.0464480i \(-0.985210\pi\)
0.998921 0.0464480i \(-0.0147902\pi\)
\(174\) 0 0
\(175\) −12470.4 + 12470.4i −0.407196 + 0.407196i
\(176\) 9608.25 + 9608.25i 0.310184 + 0.310184i
\(177\) 0 0
\(178\) −74102.0 −2.33878
\(179\) 2972.36i 0.0927674i 0.998924 + 0.0463837i \(0.0147697\pi\)
−0.998924 + 0.0463837i \(0.985230\pi\)
\(180\) 0 0
\(181\) 44034.0i 1.34410i 0.740507 + 0.672049i \(0.234586\pi\)
−0.740507 + 0.672049i \(0.765414\pi\)
\(182\) 13741.5 + 20661.5i 0.414852 + 0.623764i
\(183\) 0 0
\(184\) −9231.68 + 9231.68i −0.272675 + 0.272675i
\(185\) 79927.7 2.33536
\(186\) 0 0
\(187\) 11586.0 11586.0i 0.331323 0.331323i
\(188\) −728.122 + 728.122i −0.0206010 + 0.0206010i
\(189\) 0 0
\(190\) 105866. + 105866.i 2.93258 + 2.93258i
\(191\) 3688.96 0.101120 0.0505601 0.998721i \(-0.483899\pi\)
0.0505601 + 0.998721i \(0.483899\pi\)
\(192\) 0 0
\(193\) 19216.9 + 19216.9i 0.515904 + 0.515904i 0.916329 0.400426i \(-0.131138\pi\)
−0.400426 + 0.916329i \(0.631138\pi\)
\(194\) 24955.7i 0.663079i
\(195\) 0 0
\(196\) −39226.2 −1.02109
\(197\) 15917.0 15917.0i 0.410136 0.410136i −0.471650 0.881786i \(-0.656341\pi\)
0.881786 + 0.471650i \(0.156341\pi\)
\(198\) 0 0
\(199\) 23311.0i 0.588646i −0.955706 0.294323i \(-0.904906\pi\)
0.955706 0.294323i \(-0.0950941\pi\)
\(200\) −17457.7 + 17457.7i −0.436443 + 0.436443i
\(201\) 0 0
\(202\) 24220.3 + 24220.3i 0.593576 + 0.593576i
\(203\) −13021.8 13021.8i −0.315995 0.315995i
\(204\) 0 0
\(205\) 41884.4i 0.996654i
\(206\) 29407.0 + 29407.0i 0.692974 + 0.692974i
\(207\) 0 0
\(208\) −12933.1 19446.0i −0.298935 0.449474i
\(209\) −65193.5 −1.49249
\(210\) 0 0
\(211\) 10045.5 0.225636 0.112818 0.993616i \(-0.464012\pi\)
0.112818 + 0.993616i \(0.464012\pi\)
\(212\) 67448.9i 1.50073i
\(213\) 0 0
\(214\) 36634.6 36634.6i 0.799953 0.799953i
\(215\) 72989.0 + 72989.0i 1.57899 + 1.57899i
\(216\) 0 0
\(217\) 2268.08 0.0481658
\(218\) 104575.i 2.20047i
\(219\) 0 0
\(220\) 77915.0i 1.60981i
\(221\) −23448.8 + 15595.3i −0.480106 + 0.319308i
\(222\) 0 0
\(223\) 24945.0 24945.0i 0.501620 0.501620i −0.410321 0.911941i \(-0.634583\pi\)
0.911941 + 0.410321i \(0.134583\pi\)
\(224\) −33172.7 −0.661126
\(225\) 0 0
\(226\) −64349.9 + 64349.9i −1.25989 + 1.25989i
\(227\) −15576.9 + 15576.9i −0.302294 + 0.302294i −0.841911 0.539617i \(-0.818569\pi\)
0.539617 + 0.841911i \(0.318569\pi\)
\(228\) 0 0
\(229\) −24930.7 24930.7i −0.475405 0.475405i 0.428254 0.903658i \(-0.359129\pi\)
−0.903658 + 0.428254i \(0.859129\pi\)
\(230\) −87811.5 −1.65995
\(231\) 0 0
\(232\) −18229.7 18229.7i −0.338691 0.338691i
\(233\) 22434.3i 0.413239i −0.978421 0.206619i \(-0.933754\pi\)
0.978421 0.206619i \(-0.0662462\pi\)
\(234\) 0 0
\(235\) −1767.82 −0.0320113
\(236\) 85057.3 85057.3i 1.52717 1.52717i
\(237\) 0 0
\(238\) 24466.7i 0.431938i
\(239\) 66829.7 66829.7i 1.16997 1.16997i 0.187751 0.982217i \(-0.439880\pi\)
0.982217 0.187751i \(-0.0601199\pi\)
\(240\) 0 0
\(241\) −34827.1 34827.1i −0.599629 0.599629i 0.340585 0.940214i \(-0.389375\pi\)
−0.940214 + 0.340585i \(0.889375\pi\)
\(242\) −21526.2 21526.2i −0.367567 0.367567i
\(243\) 0 0
\(244\) 10832.7i 0.181952i
\(245\) −47619.1 47619.1i −0.793321 0.793321i
\(246\) 0 0
\(247\) 109849. + 22095.5i 1.80054 + 0.362168i
\(248\) 3175.16 0.0516253
\(249\) 0 0
\(250\) −24923.6 −0.398778
\(251\) 4768.38i 0.0756873i 0.999284 + 0.0378437i \(0.0120489\pi\)
−0.999284 + 0.0378437i \(0.987951\pi\)
\(252\) 0 0
\(253\) 27037.7 27037.7i 0.422404 0.422404i
\(254\) 47601.9 + 47601.9i 0.737832 + 0.737832i
\(255\) 0 0
\(256\) 1061.74 0.0162008
\(257\) 62743.6i 0.949956i −0.879998 0.474978i \(-0.842456\pi\)
0.879998 0.474978i \(-0.157544\pi\)
\(258\) 0 0
\(259\) 51970.3i 0.774740i
\(260\) −26407.1 + 131284.i −0.390638 + 1.94207i
\(261\) 0 0
\(262\) −60709.4 + 60709.4i −0.884409 + 0.884409i
\(263\) −45421.0 −0.656667 −0.328333 0.944562i \(-0.606487\pi\)
−0.328333 + 0.944562i \(0.606487\pi\)
\(264\) 0 0
\(265\) 81880.4 81880.4i 1.16597 1.16597i
\(266\) 68835.8 68835.8i 0.972862 0.972862i
\(267\) 0 0
\(268\) −90629.6 90629.6i −1.26183 1.26183i
\(269\) −79205.8 −1.09459 −0.547296 0.836939i \(-0.684343\pi\)
−0.547296 + 0.836939i \(0.684343\pi\)
\(270\) 0 0
\(271\) −16180.5 16180.5i −0.220320 0.220320i 0.588313 0.808633i \(-0.299792\pi\)
−0.808633 + 0.588313i \(0.799792\pi\)
\(272\) 23027.3i 0.311247i
\(273\) 0 0
\(274\) −179076. −2.38526
\(275\) 51130.0 51130.0i 0.676099 0.676099i
\(276\) 0 0
\(277\) 2884.24i 0.0375899i −0.999823 0.0187950i \(-0.994017\pi\)
0.999823 0.0187950i \(-0.00598298\pi\)
\(278\) −54478.9 + 54478.9i −0.704918 + 0.704918i
\(279\) 0 0
\(280\) 20998.9 + 20998.9i 0.267843 + 0.267843i
\(281\) 103231. + 103231.i 1.30736 + 1.30736i 0.923311 + 0.384052i \(0.125472\pi\)
0.384052 + 0.923311i \(0.374528\pi\)
\(282\) 0 0
\(283\) 78581.6i 0.981178i 0.871391 + 0.490589i \(0.163218\pi\)
−0.871391 + 0.490589i \(0.836782\pi\)
\(284\) −65006.6 65006.6i −0.805973 0.805973i
\(285\) 0 0
\(286\) −56341.9 84714.7i −0.688810 1.03568i
\(287\) 27233.9 0.330633
\(288\) 0 0
\(289\) 55753.7 0.667541
\(290\) 173400.i 2.06183i
\(291\) 0 0
\(292\) 66362.7 66362.7i 0.778320 0.778320i
\(293\) 5490.45 + 5490.45i 0.0639547 + 0.0639547i 0.738361 0.674406i \(-0.235600\pi\)
−0.674406 + 0.738361i \(0.735600\pi\)
\(294\) 0 0
\(295\) 206513. 2.37303
\(296\) 72755.0i 0.830385i
\(297\) 0 0
\(298\) 11405.0i 0.128429i
\(299\) −54721.2 + 36393.9i −0.612087 + 0.407086i
\(300\) 0 0
\(301\) 47458.7 47458.7i 0.523821 0.523821i
\(302\) −51976.9 −0.569897
\(303\) 0 0
\(304\) −64786.2 + 64786.2i −0.701028 + 0.701028i
\(305\) 13150.5 13150.5i 0.141365 0.141365i
\(306\) 0 0
\(307\) 36642.5 + 36642.5i 0.388784 + 0.388784i 0.874254 0.485469i \(-0.161351\pi\)
−0.485469 + 0.874254i \(0.661351\pi\)
\(308\) −50661.6 −0.534045
\(309\) 0 0
\(310\) 15101.0 + 15101.0i 0.157139 + 0.157139i
\(311\) 57004.6i 0.589371i 0.955594 + 0.294686i \(0.0952149\pi\)
−0.955594 + 0.294686i \(0.904785\pi\)
\(312\) 0 0
\(313\) −168911. −1.72413 −0.862064 0.506799i \(-0.830829\pi\)
−0.862064 + 0.506799i \(0.830829\pi\)
\(314\) −104907. + 104907.i −1.06401 + 1.06401i
\(315\) 0 0
\(316\) 201443.i 2.01733i
\(317\) 57585.7 57585.7i 0.573055 0.573055i −0.359926 0.932981i \(-0.617198\pi\)
0.932981 + 0.359926i \(0.117198\pi\)
\(318\) 0 0
\(319\) 53390.9 + 53390.9i 0.524670 + 0.524670i
\(320\) −163201. 163201.i −1.59376 1.59376i
\(321\) 0 0
\(322\) 57096.5i 0.550678i
\(323\) 78122.0 + 78122.0i 0.748804 + 0.748804i
\(324\) 0 0
\(325\) −103481. + 68823.2i −0.979705 + 0.651580i
\(326\) −209874. −1.97481
\(327\) 0 0
\(328\) 38125.7 0.354381
\(329\) 1149.47i 0.0106195i
\(330\) 0 0
\(331\) −134787. + 134787.i −1.23025 + 1.23025i −0.266382 + 0.963868i \(0.585828\pi\)
−0.963868 + 0.266382i \(0.914172\pi\)
\(332\) 133833. + 133833.i 1.21419 + 1.21419i
\(333\) 0 0
\(334\) 150243. 1.34680
\(335\) 220042.i 1.96072i
\(336\) 0 0
\(337\) 56729.9i 0.499520i 0.968308 + 0.249760i \(0.0803517\pi\)
−0.968308 + 0.249760i \(0.919648\pi\)
\(338\) 66222.5 + 161837.i 0.579659 + 1.41659i
\(339\) 0 0
\(340\) −93366.3 + 93366.3i −0.807667 + 0.807667i
\(341\) −9299.38 −0.0799734
\(342\) 0 0
\(343\) −71678.7 + 71678.7i −0.609259 + 0.609259i
\(344\) 66439.0 66439.0i 0.561444 0.561444i
\(345\) 0 0
\(346\) −12036.4 12036.4i −0.100541 0.100541i
\(347\) 183864. 1.52699 0.763497 0.645811i \(-0.223481\pi\)
0.763497 + 0.645811i \(0.223481\pi\)
\(348\) 0 0
\(349\) −72786.9 72786.9i −0.597588 0.597588i 0.342082 0.939670i \(-0.388868\pi\)
−0.939670 + 0.342082i \(0.888868\pi\)
\(350\) 107973.i 0.881413i
\(351\) 0 0
\(352\) 136012. 1.09772
\(353\) −106752. + 106752.i −0.856695 + 0.856695i −0.990947 0.134252i \(-0.957137\pi\)
0.134252 + 0.990947i \(0.457137\pi\)
\(354\) 0 0
\(355\) 157831.i 1.25238i
\(356\) −183867. + 183867.i −1.45078 + 1.45078i
\(357\) 0 0
\(358\) 12867.9 + 12867.9i 0.100402 + 0.100402i
\(359\) 26893.9 + 26893.9i 0.208672 + 0.208672i 0.803703 0.595031i \(-0.202860\pi\)
−0.595031 + 0.803703i \(0.702860\pi\)
\(360\) 0 0
\(361\) 309264.i 2.37309i
\(362\) 190631. + 190631.i 1.45471 + 1.45471i
\(363\) 0 0
\(364\) 85363.1 + 17170.3i 0.644269 + 0.129591i
\(365\) 161123. 1.20941
\(366\) 0 0
\(367\) 6492.32 0.0482023 0.0241012 0.999710i \(-0.492328\pi\)
0.0241012 + 0.999710i \(0.492328\pi\)
\(368\) 53737.5i 0.396809i
\(369\) 0 0
\(370\) 346022. 346022.i 2.52755 2.52755i
\(371\) −53240.0 53240.0i −0.386804 0.386804i
\(372\) 0 0
\(373\) −53286.9 −0.383004 −0.191502 0.981492i \(-0.561336\pi\)
−0.191502 + 0.981492i \(0.561336\pi\)
\(374\) 100316.i 0.717180i
\(375\) 0 0
\(376\) 1609.18i 0.0113823i
\(377\) −71866.5 108057.i −0.505643 0.760276i
\(378\) 0 0
\(379\) −92136.8 + 92136.8i −0.641438 + 0.641438i −0.950909 0.309471i \(-0.899848\pi\)
0.309471 + 0.950909i \(0.399848\pi\)
\(380\) 525363. 3.63825
\(381\) 0 0
\(382\) 15970.2 15970.2i 0.109442 0.109442i
\(383\) 15911.5 15911.5i 0.108471 0.108471i −0.650788 0.759259i \(-0.725561\pi\)
0.759259 + 0.650788i \(0.225561\pi\)
\(384\) 0 0
\(385\) −61501.3 61501.3i −0.414918 0.414918i
\(386\) 166387. 1.11672
\(387\) 0 0
\(388\) −61921.5 61921.5i −0.411319 0.411319i
\(389\) 215552.i 1.42447i −0.701941 0.712236i \(-0.747683\pi\)
0.701941 0.712236i \(-0.252317\pi\)
\(390\) 0 0
\(391\) −64799.0 −0.423852
\(392\) −43345.8 + 43345.8i −0.282082 + 0.282082i
\(393\) 0 0
\(394\) 137815.i 0.887778i
\(395\) −244544. + 244544.i −1.56734 + 1.56734i
\(396\) 0 0
\(397\) 81274.7 + 81274.7i 0.515673 + 0.515673i 0.916259 0.400586i \(-0.131194\pi\)
−0.400586 + 0.916259i \(0.631194\pi\)
\(398\) −100917. 100917.i −0.637089 0.637089i
\(399\) 0 0
\(400\) 101621.i 0.635132i
\(401\) −170564. 170564.i −1.06072 1.06072i −0.998034 0.0626817i \(-0.980035\pi\)
−0.0626817 0.998034i \(-0.519965\pi\)
\(402\) 0 0
\(403\) 15669.1 + 3151.77i 0.0964795 + 0.0194063i
\(404\) 120194. 0.736409
\(405\) 0 0
\(406\) −112748. −0.683999
\(407\) 213084.i 1.28636i
\(408\) 0 0
\(409\) −92288.1 + 92288.1i −0.551695 + 0.551695i −0.926930 0.375235i \(-0.877562\pi\)
0.375235 + 0.926930i \(0.377562\pi\)
\(410\) 181325. + 181325.i 1.07867 + 1.07867i
\(411\) 0 0
\(412\) 145933. 0.859725
\(413\) 134278.i 0.787235i
\(414\) 0 0
\(415\) 324935.i 1.88669i
\(416\) −229175. 46097.4i −1.32428 0.266373i
\(417\) 0 0
\(418\) −282235. + 282235.i −1.61532 + 1.61532i
\(419\) 20127.4 0.114646 0.0573232 0.998356i \(-0.481743\pi\)
0.0573232 + 0.998356i \(0.481743\pi\)
\(420\) 0 0
\(421\) 171209. 171209.i 0.965965 0.965965i −0.0334748 0.999440i \(-0.510657\pi\)
0.999440 + 0.0334748i \(0.0106573\pi\)
\(422\) 43489.0 43489.0i 0.244205 0.244205i
\(423\) 0 0
\(424\) −74532.5 74532.5i −0.414586 0.414586i
\(425\) −122539. −0.678417
\(426\) 0 0
\(427\) −8550.68 8550.68i −0.0468970 0.0468970i
\(428\) 181800.i 0.992446i
\(429\) 0 0
\(430\) 631966. 3.41788
\(431\) −127549. + 127549.i −0.686630 + 0.686630i −0.961485 0.274856i \(-0.911370\pi\)
0.274856 + 0.961485i \(0.411370\pi\)
\(432\) 0 0
\(433\) 68957.0i 0.367792i 0.982946 + 0.183896i \(0.0588710\pi\)
−0.982946 + 0.183896i \(0.941129\pi\)
\(434\) 9818.93 9818.93i 0.0521297 0.0521297i
\(435\) 0 0
\(436\) 259478. + 259478.i 1.36498 + 1.36498i
\(437\) 182309. + 182309.i 0.954651 + 0.954651i
\(438\) 0 0
\(439\) 113397.i 0.588400i −0.955744 0.294200i \(-0.904947\pi\)
0.955744 0.294200i \(-0.0950531\pi\)
\(440\) −86097.7 86097.7i −0.444720 0.444720i
\(441\) 0 0
\(442\) −33999.4 + 169029.i −0.174031 + 0.865203i
\(443\) −92267.8 −0.470157 −0.235079 0.971976i \(-0.575535\pi\)
−0.235079 + 0.971976i \(0.575535\pi\)
\(444\) 0 0
\(445\) −446414. −2.25433
\(446\) 215984.i 1.08580i
\(447\) 0 0
\(448\) −106116. + 106116.i −0.528719 + 0.528719i
\(449\) 17542.3 + 17542.3i 0.0870148 + 0.0870148i 0.749274 0.662260i \(-0.230402\pi\)
−0.662260 + 0.749274i \(0.730402\pi\)
\(450\) 0 0
\(451\) −111662. −0.548976
\(452\) 319338.i 1.56305i
\(453\) 0 0
\(454\) 134871.i 0.654343i
\(455\) 82783.4 + 124472.i 0.399872 + 0.601240i
\(456\) 0 0
\(457\) 104731. 104731.i 0.501466 0.501466i −0.410428 0.911893i \(-0.634621\pi\)
0.911893 + 0.410428i \(0.134621\pi\)
\(458\) −215859. −1.02906
\(459\) 0 0
\(460\) −217883. + 217883.i −1.02969 + 1.02969i
\(461\) 12382.8 12382.8i 0.0582663 0.0582663i −0.677373 0.735640i \(-0.736882\pi\)
0.735640 + 0.677373i \(0.236882\pi\)
\(462\) 0 0
\(463\) 156318. + 156318.i 0.729199 + 0.729199i 0.970460 0.241261i \(-0.0775611\pi\)
−0.241261 + 0.970460i \(0.577561\pi\)
\(464\) 106115. 0.492878
\(465\) 0 0
\(466\) −97122.3 97122.3i −0.447247 0.447247i
\(467\) 3880.03i 0.0177911i −0.999960 0.00889553i \(-0.997168\pi\)
0.999960 0.00889553i \(-0.00283157\pi\)
\(468\) 0 0
\(469\) −143075. −0.650455
\(470\) −7653.24 + 7653.24i −0.0346457 + 0.0346457i
\(471\) 0 0
\(472\) 187980.i 0.843778i
\(473\) −194586. + 194586.i −0.869740 + 0.869740i
\(474\) 0 0
\(475\) 344758. + 344758.i 1.52801 + 1.52801i
\(476\) 60708.3 + 60708.3i 0.267938 + 0.267938i
\(477\) 0 0
\(478\) 578637.i 2.53250i
\(479\) −53446.0 53446.0i −0.232940 0.232940i 0.580979 0.813919i \(-0.302670\pi\)
−0.813919 + 0.580979i \(0.802670\pi\)
\(480\) 0 0
\(481\) 72218.9 359040.i 0.312148 1.55186i
\(482\) −301546. −1.29795
\(483\) 0 0
\(484\) −106824. −0.456015
\(485\) 150341.i 0.639136i
\(486\) 0 0
\(487\) −39739.5 + 39739.5i −0.167558 + 0.167558i −0.785905 0.618347i \(-0.787803\pi\)
0.618347 + 0.785905i \(0.287803\pi\)
\(488\) −11970.4 11970.4i −0.0502653 0.0502653i
\(489\) 0 0
\(490\) −412304. −1.71722
\(491\) 183063.i 0.759342i −0.925122 0.379671i \(-0.876037\pi\)
0.925122 0.379671i \(-0.123963\pi\)
\(492\) 0 0
\(493\) 127958.i 0.526469i
\(494\) 571212. 379900.i 2.34069 1.55674i
\(495\) 0 0
\(496\) −9241.28 + 9241.28i −0.0375638 + 0.0375638i
\(497\) −102624. −0.415468
\(498\) 0 0
\(499\) −57389.5 + 57389.5i −0.230479 + 0.230479i −0.812893 0.582414i \(-0.802108\pi\)
0.582414 + 0.812893i \(0.302108\pi\)
\(500\) −61842.1 + 61842.1i −0.247368 + 0.247368i
\(501\) 0 0
\(502\) 20643.2 + 20643.2i 0.0819161 + 0.0819161i
\(503\) 192555. 0.761061 0.380530 0.924768i \(-0.375741\pi\)
0.380530 + 0.924768i \(0.375741\pi\)
\(504\) 0 0
\(505\) 145910. + 145910.i 0.572142 + 0.572142i
\(506\) 234102.i 0.914333i
\(507\) 0 0
\(508\) 236226. 0.915377
\(509\) 84764.7 84764.7i 0.327175 0.327175i −0.524336 0.851511i \(-0.675687\pi\)
0.851511 + 0.524336i \(0.175687\pi\)
\(510\) 0 0
\(511\) 104765.i 0.401213i
\(512\) 187652. 187652.i 0.715835 0.715835i
\(513\) 0 0
\(514\) −271629. 271629.i −1.02813 1.02813i
\(515\) 177157. + 177157.i 0.667951 + 0.667951i
\(516\) 0 0
\(517\) 4712.95i 0.0176324i
\(518\) −224989. 224989.i −0.838498 0.838498i
\(519\) 0 0
\(520\) 115891. + 174252.i 0.428592 + 0.644424i
\(521\) −314076. −1.15707 −0.578534 0.815658i \(-0.696375\pi\)
−0.578534 + 0.815658i \(0.696375\pi\)
\(522\) 0 0
\(523\) −37022.8 −0.135352 −0.0676762 0.997707i \(-0.521558\pi\)
−0.0676762 + 0.997707i \(0.521558\pi\)
\(524\) 301272.i 1.09723i
\(525\) 0 0
\(526\) −196636. + 196636.i −0.710708 + 0.710708i
\(527\) 11143.5 + 11143.5i 0.0401238 + 0.0401238i
\(528\) 0 0
\(529\) 128624. 0.459631
\(530\) 708951.i 2.52386i
\(531\) 0 0
\(532\) 341600.i 1.20696i
\(533\) 188147. + 37844.8i 0.662282 + 0.133214i
\(534\) 0 0
\(535\) 220699. 220699.i 0.771067 0.771067i
\(536\) −200295. −0.697174
\(537\) 0 0
\(538\) −342897. + 342897.i −1.18467 + 1.18467i
\(539\) 126951. 126951.i 0.436976 0.436976i
\(540\) 0 0
\(541\) −38918.2 38918.2i −0.132971 0.132971i 0.637489 0.770460i \(-0.279973\pi\)
−0.770460 + 0.637489i \(0.779973\pi\)
\(542\) −140097. −0.476903
\(543\) 0 0
\(544\) −162984. 162984.i −0.550741 0.550741i
\(545\) 629992.i 2.12101i
\(546\) 0 0
\(547\) −383774. −1.28263 −0.641314 0.767279i \(-0.721610\pi\)
−0.641314 + 0.767279i \(0.721610\pi\)
\(548\) −444334. + 444334.i −1.47962 + 1.47962i
\(549\) 0 0
\(550\) 442702.i 1.46348i
\(551\) −360003. + 360003.i −1.18578 + 1.18578i
\(552\) 0 0
\(553\) 159007. + 159007.i 0.519954 + 0.519954i
\(554\) −12486.4 12486.4i −0.0406835 0.0406835i
\(555\) 0 0
\(556\) 270353.i 0.874543i
\(557\) −280053. 280053.i −0.902672 0.902672i 0.0929944 0.995667i \(-0.470356\pi\)
−0.995667 + 0.0929944i \(0.970356\pi\)
\(558\) 0 0
\(559\) 393820. 261921.i 1.26030 0.838199i
\(560\) −122234. −0.389777
\(561\) 0 0
\(562\) 893810. 2.82991
\(563\) 50493.5i 0.159301i −0.996823 0.0796505i \(-0.974620\pi\)
0.996823 0.0796505i \(-0.0253804\pi\)
\(564\) 0 0
\(565\) −387664. + 387664.i −1.21439 + 1.21439i
\(566\) 340194. + 340194.i 1.06193 + 1.06193i
\(567\) 0 0
\(568\) −143667. −0.445309
\(569\) 301875.i 0.932400i 0.884679 + 0.466200i \(0.154377\pi\)
−0.884679 + 0.466200i \(0.845623\pi\)
\(570\) 0 0
\(571\) 290046.i 0.889601i −0.895630 0.444800i \(-0.853275\pi\)
0.895630 0.444800i \(-0.146725\pi\)
\(572\) −349999. 70400.4i −1.06973 0.215171i
\(573\) 0 0
\(574\) 117901. 117901.i 0.357843 0.357843i
\(575\) −285962. −0.864914
\(576\) 0 0
\(577\) 127765. 127765.i 0.383760 0.383760i −0.488695 0.872455i \(-0.662527\pi\)
0.872455 + 0.488695i \(0.162527\pi\)
\(578\) 241368. 241368.i 0.722477 0.722477i
\(579\) 0 0
\(580\) −430252. 430252.i −1.27899 1.27899i
\(581\) 211278. 0.625897
\(582\) 0 0
\(583\) 218290. + 218290.i 0.642240 + 0.642240i
\(584\) 146664.i 0.430030i
\(585\) 0 0
\(586\) 47538.4 0.138436
\(587\) 432681. 432681.i 1.25572 1.25572i 0.302598 0.953118i \(-0.402146\pi\)
0.953118 0.302598i \(-0.0978541\pi\)
\(588\) 0 0
\(589\) 62703.6i 0.180743i
\(590\) 894031. 894031.i 2.56832 2.56832i
\(591\) 0 0
\(592\) 211753. + 211753.i 0.604207 + 0.604207i
\(593\) −351836. 351836.i −1.00053 1.00053i −1.00000 0.000531251i \(-0.999831\pi\)
−0.000531251 1.00000i \(-0.500169\pi\)
\(594\) 0 0
\(595\) 147395.i 0.416341i
\(596\) −28298.7 28298.7i −0.0796663 0.0796663i
\(597\) 0 0
\(598\) −79342.4 + 394454.i −0.221872 + 1.10305i
\(599\) 484810. 1.35119 0.675597 0.737271i \(-0.263886\pi\)
0.675597 + 0.737271i \(0.263886\pi\)
\(600\) 0 0
\(601\) 214570. 0.594046 0.297023 0.954870i \(-0.404006\pi\)
0.297023 + 0.954870i \(0.404006\pi\)
\(602\) 410915.i 1.13386i
\(603\) 0 0
\(604\) −128968. + 128968.i −0.353516 + 0.353516i
\(605\) −129681. 129681.i −0.354294 0.354294i
\(606\) 0 0
\(607\) −2800.08 −0.00759964 −0.00379982 0.999993i \(-0.501210\pi\)
−0.00379982 + 0.999993i \(0.501210\pi\)
\(608\) 917096.i 2.48089i
\(609\) 0 0
\(610\) 113862.i 0.305998i
\(611\) −1597.32 + 7941.17i −0.00427869 + 0.0212717i
\(612\) 0 0
\(613\) −92722.6 + 92722.6i −0.246754 + 0.246754i −0.819637 0.572883i \(-0.805825\pi\)
0.572883 + 0.819637i \(0.305825\pi\)
\(614\) 317265. 0.841560
\(615\) 0 0
\(616\) −55982.2 + 55982.2i −0.147533 + 0.147533i
\(617\) −260927. + 260927.i −0.685407 + 0.685407i −0.961213 0.275806i \(-0.911055\pi\)
0.275806 + 0.961213i \(0.411055\pi\)
\(618\) 0 0
\(619\) 103595. + 103595.i 0.270369 + 0.270369i 0.829249 0.558880i \(-0.188769\pi\)
−0.558880 + 0.829249i \(0.688769\pi\)
\(620\) 74939.2 0.194951
\(621\) 0 0
\(622\) 246783. + 246783.i 0.637874 + 0.637874i
\(623\) 290266.i 0.747859i
\(624\) 0 0
\(625\) 309460. 0.792218
\(626\) −731248. + 731248.i −1.86602 + 1.86602i
\(627\) 0 0
\(628\) 520606.i 1.32005i
\(629\) 255341. 255341.i 0.645385 0.645385i
\(630\) 0 0
\(631\) −300176. 300176.i −0.753907 0.753907i 0.221299 0.975206i \(-0.428970\pi\)
−0.975206 + 0.221299i \(0.928970\pi\)
\(632\) 222599. + 222599.i 0.557299 + 0.557299i
\(633\) 0 0
\(634\) 498599.i 1.24043i
\(635\) 286769. + 286769.i 0.711189 + 0.711189i
\(636\) 0 0
\(637\) −256934. + 170881.i −0.633203 + 0.421130i
\(638\) 462279. 1.13570
\(639\) 0 0
\(640\) −596770. −1.45696
\(641\) 466494.i 1.13535i −0.823253 0.567675i \(-0.807843\pi\)
0.823253 0.567675i \(-0.192157\pi\)
\(642\) 0 0
\(643\) 439463. 439463.i 1.06292 1.06292i 0.0650372 0.997883i \(-0.479283\pi\)
0.997883 0.0650372i \(-0.0207166\pi\)
\(644\) 141671. + 141671.i 0.341594 + 0.341594i
\(645\) 0 0
\(646\) 676409. 1.62086
\(647\) 562437.i 1.34358i 0.740740 + 0.671792i \(0.234475\pi\)
−0.740740 + 0.671792i \(0.765525\pi\)
\(648\) 0 0
\(649\) 550555.i 1.30711i
\(650\) −150042. + 745938.i −0.355128 + 1.76553i
\(651\) 0 0
\(652\) −520754. + 520754.i −1.22500 + 1.22500i
\(653\) −37629.2 −0.0882467 −0.0441234 0.999026i \(-0.514049\pi\)
−0.0441234 + 0.999026i \(0.514049\pi\)
\(654\) 0 0
\(655\) −365733. + 365733.i −0.852474 + 0.852474i
\(656\) −110965. + 110965.i −0.257856 + 0.257856i
\(657\) 0 0
\(658\) 4976.27 + 4976.27i 0.0114935 + 0.0114935i
\(659\) −18033.4 −0.0415248 −0.0207624 0.999784i \(-0.506609\pi\)
−0.0207624 + 0.999784i \(0.506609\pi\)
\(660\) 0 0
\(661\) 536607. + 536607.i 1.22816 + 1.22816i 0.964661 + 0.263495i \(0.0848754\pi\)
0.263495 + 0.964661i \(0.415125\pi\)
\(662\) 1.16704e6i 2.66299i
\(663\) 0 0
\(664\) 295776. 0.670852
\(665\) 414689. 414689.i 0.937733 0.937733i
\(666\) 0 0
\(667\) 298607.i 0.671195i
\(668\) 372794. 372794.i 0.835440 0.835440i
\(669\) 0 0
\(670\) −952601. 952601.i −2.12208 2.12208i
\(671\) 35058.7 + 35058.7i 0.0778666 + 0.0778666i
\(672\) 0 0
\(673\) 75975.6i 0.167743i 0.996477 + 0.0838715i \(0.0267285\pi\)
−0.996477 + 0.0838715i \(0.973271\pi\)
\(674\) 245594. + 245594.i 0.540628 + 0.540628i
\(675\) 0 0
\(676\) 565875. + 237244.i 1.23830 + 0.519162i
\(677\) −438381. −0.956478 −0.478239 0.878230i \(-0.658725\pi\)
−0.478239 + 0.878230i \(0.658725\pi\)
\(678\) 0 0
\(679\) −97754.1 −0.212029
\(680\) 206343.i 0.446244i
\(681\) 0 0
\(682\) −40258.7 + 40258.7i −0.0865549 + 0.0865549i
\(683\) 14323.9 + 14323.9i 0.0307057 + 0.0307057i 0.722293 0.691587i \(-0.243088\pi\)
−0.691587 + 0.722293i \(0.743088\pi\)
\(684\) 0 0
\(685\) −1.07881e6 −2.29913
\(686\) 620620.i 1.31880i
\(687\) 0 0
\(688\) 386741.i 0.817039i
\(689\) −293828. 441795.i −0.618949 0.930641i
\(690\) 0 0
\(691\) 64315.1 64315.1i 0.134697 0.134697i −0.636544 0.771241i \(-0.719637\pi\)
0.771241 + 0.636544i \(0.219637\pi\)
\(692\) −59730.8 −0.124734
\(693\) 0 0
\(694\) 795981. 795981.i 1.65266 1.65266i
\(695\) −328198. + 328198.i −0.679464 + 0.679464i
\(696\) 0 0
\(697\) 133806. + 133806.i 0.275429 + 0.275429i
\(698\) −630216. −1.29354
\(699\) 0 0
\(700\) 267910. + 267910.i 0.546755 + 0.546755i
\(701\) 77204.1i 0.157110i −0.996910 0.0785551i \(-0.974969\pi\)
0.996910 0.0785551i \(-0.0250307\pi\)
\(702\) 0 0
\(703\) −1.43678e6 −2.90723
\(704\) 435088. 435088.i 0.877873 0.877873i
\(705\) 0 0
\(706\) 924298.i 1.85440i
\(707\) 94873.4 94873.4i 0.189804 0.189804i
\(708\) 0 0
\(709\) 561910. + 561910.i 1.11783 + 1.11783i 0.992060 + 0.125766i \(0.0401389\pi\)
0.125766 + 0.992060i \(0.459861\pi\)
\(710\) −683279. 683279.i −1.35544 1.35544i
\(711\) 0 0
\(712\) 406353.i 0.801574i
\(713\) 26005.0 + 26005.0i 0.0511538 + 0.0511538i
\(714\) 0 0
\(715\) −339421. 510348.i −0.663937 0.998285i
\(716\) 63857.3 0.124562
\(717\) 0 0
\(718\) 232857. 0.451690
\(719\) 324777.i 0.628243i 0.949383 + 0.314121i \(0.101710\pi\)
−0.949383 + 0.314121i \(0.898290\pi\)
\(720\) 0 0
\(721\) 115191. 115191.i 0.221588 0.221588i
\(722\) −1.33886e6 1.33886e6i −2.56839 2.56839i
\(723\) 0 0
\(724\) 946013. 1.80476
\(725\) 564686.i 1.07431i
\(726\) 0 0
\(727\) 922269.i 1.74497i −0.488638 0.872487i \(-0.662506\pi\)
0.488638 0.872487i \(-0.337494\pi\)
\(728\) 113302. 75354.5i 0.213783 0.142183i
\(729\) 0 0
\(730\) 697533. 697533.i 1.30894 1.30894i
\(731\) 466348. 0.872722
\(732\) 0 0
\(733\) 295621. 295621.i 0.550209 0.550209i −0.376292 0.926501i \(-0.622801\pi\)
0.926501 + 0.376292i \(0.122801\pi\)
\(734\) 28106.5 28106.5i 0.0521692 0.0521692i
\(735\) 0 0
\(736\) −380346. 380346.i −0.702140 0.702140i
\(737\) 586623. 1.08000
\(738\) 0 0
\(739\) 369186. + 369186.i 0.676015 + 0.676015i 0.959096 0.283081i \(-0.0913566\pi\)
−0.283081 + 0.959096i \(0.591357\pi\)
\(740\) 1.71714e6i 3.13576i
\(741\) 0 0
\(742\) −460972. −0.837272
\(743\) 671107. 671107.i 1.21567 1.21567i 0.246530 0.969135i \(-0.420710\pi\)
0.969135 0.246530i \(-0.0792905\pi\)
\(744\) 0 0
\(745\) 68707.2i 0.123791i
\(746\) −230689. + 230689.i −0.414523 + 0.414523i
\(747\) 0 0
\(748\) −248911. 248911.i −0.444878 0.444878i
\(749\) −143502. 143502.i −0.255796 0.255796i
\(750\) 0 0
\(751\) 167627.i 0.297210i 0.988897 + 0.148605i \(0.0474784\pi\)
−0.988897 + 0.148605i \(0.952522\pi\)
\(752\) −4683.51 4683.51i −0.00828201 0.00828201i
\(753\) 0 0
\(754\) −778924. 156676.i −1.37010 0.275588i
\(755\) −313125. −0.549319
\(756\) 0 0
\(757\) −578076. −1.00877 −0.504386 0.863478i \(-0.668281\pi\)
−0.504386 + 0.863478i \(0.668281\pi\)
\(758\) 797754.i 1.38845i
\(759\) 0 0
\(760\) 580537. 580537.i 1.00508 1.00508i
\(761\) 762616. + 762616.i 1.31685 + 1.31685i 0.916254 + 0.400597i \(0.131197\pi\)
0.400597 + 0.916254i \(0.368803\pi\)
\(762\) 0 0
\(763\) 409632. 0.703630
\(764\) 79252.6i 0.135777i
\(765\) 0 0
\(766\) 137768.i 0.234795i
\(767\) 186595. 927666.i 0.317183 1.57689i
\(768\) 0 0
\(769\) 698760. 698760.i 1.18161 1.18161i 0.202287 0.979326i \(-0.435163\pi\)
0.979326 0.202287i \(-0.0648374\pi\)
\(770\) −532501. −0.898129
\(771\) 0 0
\(772\) 412850. 412850.i 0.692720 0.692720i
\(773\) 311294. 311294.i 0.520968 0.520968i −0.396895 0.917864i \(-0.629912\pi\)
0.917864 + 0.396895i \(0.129912\pi\)
\(774\) 0 0
\(775\) 49177.2 + 49177.2i 0.0818766 + 0.0818766i
\(776\) −136849. −0.227258
\(777\) 0 0
\(778\) −933166. 933166.i −1.54170 1.54170i
\(779\) 752912.i 1.24071i
\(780\) 0 0
\(781\) 420771. 0.689833
\(782\) −280527. + 280527.i −0.458734 + 0.458734i
\(783\) 0 0
\(784\) 252315.i 0.410498i
\(785\) −631996. + 631996.i −1.02559 + 1.02559i
\(786\) 0 0
\(787\) −528223. 528223.i −0.852841 0.852841i 0.137642 0.990482i \(-0.456048\pi\)
−0.990482 + 0.137642i \(0.956048\pi\)
\(788\) −341955. 341955.i −0.550702 0.550702i
\(789\) 0 0
\(790\) 2.11735e6i 3.39265i
\(791\) 252066. + 252066.i 0.402866 + 0.402866i
\(792\) 0 0
\(793\) −47190.6 70955.0i −0.0750428 0.112833i
\(794\) 703706. 1.11622
\(795\) 0 0
\(796\) −500805. −0.790393
\(797\) 550495.i 0.866636i 0.901241 + 0.433318i \(0.142657\pi\)
−0.901241 + 0.433318i \(0.857343\pi\)
\(798\) 0 0
\(799\) −5647.58 + 5647.58i −0.00884644 + 0.00884644i
\(800\) −719260. 719260.i −1.12384 1.12384i
\(801\) 0 0
\(802\) −1.47681e6 −2.29602
\(803\) 429549.i 0.666165i
\(804\) 0 0
\(805\) 343967.i 0.530793i
\(806\) 81479.2 54190.0i 0.125423 0.0834160i
\(807\) 0 0
\(808\) 132817. 132817.i 0.203437 0.203437i
\(809\) −705446. −1.07787 −0.538936 0.842347i \(-0.681173\pi\)
−0.538936 + 0.842347i \(0.681173\pi\)
\(810\) 0 0
\(811\) 476089. 476089.i 0.723846 0.723846i −0.245540 0.969386i \(-0.578965\pi\)
0.969386 + 0.245540i \(0.0789654\pi\)
\(812\) −279757. + 279757.i −0.424296 + 0.424296i
\(813\) 0 0
\(814\) 922481. + 922481.i 1.39222 + 1.39222i
\(815\) −1.26435e6 −1.90350
\(816\) 0 0
\(817\) −1.31205e6 1.31205e6i −1.96565 1.96565i
\(818\) 799064.i 1.19419i
\(819\) 0 0
\(820\) 899831. 1.33824
\(821\) −162224. + 162224.i −0.240675 + 0.240675i −0.817129 0.576455i \(-0.804436\pi\)
0.576455 + 0.817129i \(0.304436\pi\)
\(822\) 0 0
\(823\) 1.34448e6i 1.98497i 0.122378 + 0.992484i \(0.460948\pi\)
−0.122378 + 0.992484i \(0.539052\pi\)
\(824\) 161259. 161259.i 0.237504 0.237504i
\(825\) 0 0
\(826\) −581314. 581314.i −0.852022 0.852022i
\(827\) 542193. + 542193.i 0.792762 + 0.792762i 0.981942 0.189180i \(-0.0605830\pi\)
−0.189180 + 0.981942i \(0.560583\pi\)
\(828\) 0 0
\(829\) 485612.i 0.706611i 0.935508 + 0.353305i \(0.114942\pi\)
−0.935508 + 0.353305i \(0.885058\pi\)
\(830\) 1.40671e6 + 1.40671e6i 2.04196 + 2.04196i
\(831\) 0 0
\(832\) −880569. + 585648.i −1.27209 + 0.846038i
\(833\) −304253. −0.438474
\(834\) 0 0
\(835\) 905114. 1.29817
\(836\) 1.40060e6i 2.00401i
\(837\) 0 0
\(838\) 87135.4 87135.4i 0.124081 0.124081i
\(839\) −702797. 702797.i −0.998404 0.998404i 0.00159522 0.999999i \(-0.499492\pi\)
−0.999999 + 0.00159522i \(0.999492\pi\)
\(840\) 0 0
\(841\) −117625. −0.166305
\(842\) 1.48239e6i 2.09092i
\(843\) 0 0
\(844\) 215815.i 0.302969i
\(845\) 398945. + 974957.i 0.558727 + 1.36544i
\(846\) 0 0
\(847\) −84320.5 + 84320.5i −0.117535 + 0.117535i
\(848\) 433853. 0.603324
\(849\) 0 0
\(850\) −530494. + 530494.i −0.734248 + 0.734248i
\(851\) 595874. 595874.i 0.822801 0.822801i
\(852\) 0 0
\(853\) 528758. + 528758.i 0.726706 + 0.726706i 0.969962 0.243256i \(-0.0782154\pi\)
−0.243256 + 0.969962i \(0.578215\pi\)
\(854\) −74034.9 −0.101513
\(855\) 0 0
\(856\) −200893. 200893.i −0.274169 0.274169i
\(857\) 427769.i 0.582435i −0.956657 0.291217i \(-0.905940\pi\)
0.956657 0.291217i \(-0.0940603\pi\)
\(858\) 0 0
\(859\) −110526. −0.149788 −0.0748941 0.997191i \(-0.523862\pi\)
−0.0748941 + 0.997191i \(0.523862\pi\)
\(860\) 1.56807e6 1.56807e6i 2.12016 2.12016i
\(861\) 0 0
\(862\) 1.10437e6i 1.48627i
\(863\) −115225. + 115225.i −0.154713 + 0.154713i −0.780219 0.625506i \(-0.784892\pi\)
0.625506 + 0.780219i \(0.284892\pi\)
\(864\) 0 0
\(865\) −72510.9 72510.9i −0.0969106 0.0969106i
\(866\) 298528. + 298528.i 0.398060 + 0.398060i
\(867\) 0 0
\(868\) 48726.7i 0.0646737i
\(869\) −651945. 651945.i −0.863319 0.863319i
\(870\) 0 0
\(871\) −988440. 198819.i −1.30291 0.262073i
\(872\) 573457. 0.754168
\(873\) 0 0
\(874\) 1.57850e6 2.06643
\(875\) 97628.6i 0.127515i
\(876\) 0 0
\(877\) 352595. 352595.i 0.458435 0.458435i −0.439707 0.898141i \(-0.644918\pi\)
0.898141 + 0.439707i \(0.144918\pi\)
\(878\) −490917. 490917.i −0.636823 0.636823i
\(879\) 0 0
\(880\) 501174. 0.647177
\(881\) 776892.i 1.00094i −0.865753 0.500471i \(-0.833160\pi\)
0.865753 0.500471i \(-0.166840\pi\)
\(882\) 0 0
\(883\) 990585.i 1.27049i 0.772312 + 0.635244i \(0.219100\pi\)
−0.772312 + 0.635244i \(0.780900\pi\)
\(884\) 335045. + 503768.i 0.428745 + 0.644653i
\(885\) 0 0
\(886\) −399445. + 399445.i −0.508849 + 0.508849i
\(887\) −951592. −1.20949 −0.604747 0.796418i \(-0.706726\pi\)
−0.604747 + 0.796418i \(0.706726\pi\)
\(888\) 0 0
\(889\) 186462. 186462.i 0.235932 0.235932i
\(890\) −1.93261e6 + 1.93261e6i −2.43985 + 2.43985i
\(891\) 0 0
\(892\) −535912. 535912.i −0.673540 0.673540i
\(893\) 31778.4 0.0398500
\(894\) 0 0
\(895\) 77520.3 + 77520.3i 0.0967764 + 0.0967764i
\(896\) 388030.i 0.483336i
\(897\) 0 0
\(898\) 151887. 0.188352
\(899\) −51351.8 + 51351.8i −0.0635384 + 0.0635384i
\(900\) 0 0
\(901\) 523158.i 0.644442i
\(902\) −483406. + 483406.i −0.594154 + 0.594154i
\(903\) 0 0
\(904\) 352875. + 352875.i 0.431802 + 0.431802i
\(905\) 1.14842e6 + 1.14842e6i 1.40218 + 1.40218i
\(906\) 0 0
\(907\) 122762.i 0.149228i −0.997212 0.0746141i \(-0.976227\pi\)
0.997212 0.0746141i \(-0.0237725\pi\)
\(908\) 334650. + 334650.i 0.405900 + 0.405900i
\(909\) 0 0
\(910\) 897246. + 180476.i 1.08350 + 0.217940i
\(911\) −93939.2 −0.113191 −0.0565953 0.998397i \(-0.518024\pi\)
−0.0565953 + 0.998397i \(0.518024\pi\)
\(912\) 0 0
\(913\) −866266. −1.03923
\(914\) 906796.i 1.08547i
\(915\) 0 0
\(916\) −535604. + 535604.i −0.638341 + 0.638341i
\(917\) 237805. + 237805.i 0.282802 + 0.282802i
\(918\) 0 0
\(919\) 217922. 0.258030 0.129015 0.991643i \(-0.458818\pi\)
0.129015 + 0.991643i \(0.458818\pi\)
\(920\) 481531.i 0.568917i
\(921\) 0 0
\(922\) 107215.i 0.126123i
\(923\) −708986. 142609.i −0.832212 0.167395i
\(924\) 0 0
\(925\) 1.12684e6 1.12684e6i 1.31697 1.31697i
\(926\) 1.35346e6 1.57842
\(927\) 0 0
\(928\) 751065. 751065.i 0.872131 0.872131i
\(929\) 489424. 489424.i 0.567092 0.567092i −0.364221 0.931313i \(-0.618665\pi\)
0.931313 + 0.364221i \(0.118665\pi\)
\(930\) 0 0
\(931\) 856000. + 856000.i 0.987584 + 0.987584i
\(932\) −481972. −0.554868
\(933\) 0 0
\(934\) −16797.4 16797.4i −0.0192552 0.0192552i
\(935\) 604337.i 0.691283i
\(936\) 0 0
\(937\) 74994.8 0.0854185 0.0427093 0.999088i \(-0.486401\pi\)
0.0427093 + 0.999088i \(0.486401\pi\)
\(938\) −619397. + 619397.i −0.703985 + 0.703985i
\(939\) 0 0
\(940\) 37979.4i 0.0429826i
\(941\) 746527. 746527.i 0.843075 0.843075i −0.146183 0.989258i \(-0.546699\pi\)
0.989258 + 0.146183i \(0.0466988\pi\)
\(942\) 0 0
\(943\) 312255. + 312255.i 0.351144 + 0.351144i
\(944\) 547115. + 547115.i 0.613953 + 0.613953i
\(945\) 0 0
\(946\) 1.68480e6i 1.88263i
\(947\) 392710. + 392710.i 0.437897 + 0.437897i 0.891304 0.453407i \(-0.149792\pi\)
−0.453407 + 0.891304i \(0.649792\pi\)
\(948\) 0 0
\(949\) 145584. 723776.i 0.161652 0.803659i
\(950\) 2.98504e6 3.30752
\(951\) 0 0
\(952\) 134168. 0.148039
\(953\) 920738.i 1.01380i 0.862006 + 0.506898i \(0.169208\pi\)
−0.862006 + 0.506898i \(0.830792\pi\)
\(954\) 0 0
\(955\) 96209.5 96209.5i 0.105490 0.105490i
\(956\) −1.43575e6 1.43575e6i −1.57095 1.57095i
\(957\) 0 0
\(958\) −462756. −0.504221
\(959\) 701460.i 0.762721i
\(960\) 0 0
\(961\) 914577.i 0.990315i
\(962\) −1.24170e6 1.86700e6i −1.34173 2.01741i
\(963\) 0 0
\(964\) −748214. + 748214.i −0.805141 + 0.805141i
\(965\) 1.00237e6 1.07640
\(966\) 0 0
\(967\) −683545. + 683545.i −0.730994 + 0.730994i −0.970817 0.239823i \(-0.922911\pi\)
0.239823 + 0.970817i \(0.422911\pi\)
\(968\) −118043. + 118043.i −0.125977 + 0.125977i
\(969\) 0 0
\(970\) −650853. 650853.i −0.691734 0.691734i
\(971\) 107226. 0.113726 0.0568632 0.998382i \(-0.481890\pi\)
0.0568632 + 0.998382i \(0.481890\pi\)
\(972\) 0 0
\(973\) 213400. + 213400.i 0.225407 + 0.225407i
\(974\) 344080.i 0.362695i
\(975\) 0 0
\(976\) 69679.4 0.0731484
\(977\) 730243. 730243.i 0.765030 0.765030i −0.212197 0.977227i \(-0.568062\pi\)
0.977227 + 0.212197i \(0.0680619\pi\)
\(978\) 0 0
\(979\) 1.19012e6i 1.24173i
\(980\) −1.02303e6 + 1.02303e6i −1.06522 + 1.06522i
\(981\) 0 0
\(982\) −792514. 792514.i −0.821833 0.821833i
\(983\) −180212. 180212.i −0.186500 0.186500i 0.607681 0.794181i \(-0.292100\pi\)
−0.794181 + 0.607681i \(0.792100\pi\)
\(984\) 0 0
\(985\) 830241.i 0.855720i
\(986\) −553953. 553953.i −0.569795 0.569795i
\(987\) 0 0
\(988\) 474693. 2.35996e6i 0.486294 2.41763i
\(989\) 1.08829e6 1.11263
\(990\) 0 0
\(991\) −662719. −0.674811 −0.337405 0.941359i \(-0.609549\pi\)
−0.337405 + 0.941359i \(0.609549\pi\)
\(992\) 130817.i 0.132936i
\(993\) 0 0
\(994\) −444280. + 444280.i −0.449659 + 0.449659i
\(995\) −607959. 607959.i −0.614084 0.614084i
\(996\) 0 0
\(997\) 464482. 0.467281 0.233641 0.972323i \(-0.424936\pi\)
0.233641 + 0.972323i \(0.424936\pi\)
\(998\) 496899.i 0.498893i
\(999\) 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 117.5.j.b.109.9 20
3.2 odd 2 39.5.g.a.31.2 20
13.8 odd 4 inner 117.5.j.b.73.9 20
39.8 even 4 39.5.g.a.34.2 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.5.g.a.31.2 20 3.2 odd 2
39.5.g.a.34.2 yes 20 39.8 even 4
117.5.j.b.73.9 20 13.8 odd 4 inner
117.5.j.b.109.9 20 1.1 even 1 trivial