Properties

Label 108.5.d.b.55.7
Level $108$
Weight $5$
Character 108.55
Analytic conductor $11.164$
Analytic rank $0$
Dimension $16$
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] = [108,5,Mod(55,108)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(108, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("108.55");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 108.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.1639560131\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 38x^{14} + 1016x^{12} + 13512x^{10} + 130640x^{8} + 569472x^{6} + 1783808x^{4} + 352256x^{2} + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{24} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 55.7
Root \(-2.23772 - 3.87585i\) of defining polynomial
Character \(\chi\) \(=\) 108.55
Dual form 108.5.d.b.55.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.988828 - 3.87585i) q^{2} +(-14.0444 + 7.66510i) q^{4} +20.7568 q^{5} -5.38785i q^{7} +(43.5963 + 46.8547i) q^{8} +O(q^{10})\) \(q+(-0.988828 - 3.87585i) q^{2} +(-14.0444 + 7.66510i) q^{4} +20.7568 q^{5} -5.38785i q^{7} +(43.5963 + 46.8547i) q^{8} +(-20.5250 - 80.4504i) q^{10} +115.619i q^{11} +207.234 q^{13} +(-20.8825 + 5.32765i) q^{14} +(138.493 - 215.304i) q^{16} +383.437 q^{17} -618.390i q^{19} +(-291.518 + 159.103i) q^{20} +(448.121 - 114.327i) q^{22} -82.9899i q^{23} -194.153 q^{25} +(-204.919 - 803.209i) q^{26} +(41.2984 + 75.6693i) q^{28} +201.949 q^{29} -196.140i q^{31} +(-971.432 - 323.878i) q^{32} +(-379.153 - 1486.14i) q^{34} -111.835i q^{35} +340.480 q^{37} +(-2396.79 + 611.481i) q^{38} +(904.922 + 972.556i) q^{40} +2797.22 q^{41} -254.774i q^{43} +(-886.230 - 1623.80i) q^{44} +(-321.656 + 82.0627i) q^{46} +2257.41i q^{47} +2371.97 q^{49} +(191.984 + 752.509i) q^{50} +(-2910.49 + 1588.47i) q^{52} +4111.23 q^{53} +2399.88i q^{55} +(252.446 - 234.890i) q^{56} +(-199.693 - 782.726i) q^{58} -1598.57i q^{59} -6081.09 q^{61} +(-760.210 + 193.949i) q^{62} +(-294.723 + 4085.38i) q^{64} +4301.53 q^{65} +7383.67i q^{67} +(-5385.15 + 2939.08i) q^{68} +(-433.455 + 110.585i) q^{70} +9347.39i q^{71} -5320.95 q^{73} +(-336.676 - 1319.65i) q^{74} +(4740.02 + 8684.94i) q^{76} +622.936 q^{77} -5971.42i q^{79} +(2874.67 - 4469.03i) q^{80} +(-2765.97 - 10841.6i) q^{82} -10091.0i q^{83} +7958.94 q^{85} +(-987.466 + 251.928i) q^{86} +(-5417.28 + 5040.55i) q^{88} -13611.5 q^{89} -1116.55i q^{91} +(636.126 + 1165.55i) q^{92} +(8749.37 - 2232.19i) q^{94} -12835.8i q^{95} +2179.54 q^{97} +(-2345.47 - 9193.41i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 28 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 28 q^{4} + 176 q^{10} + 176 q^{13} + 88 q^{16} + 384 q^{22} + 2736 q^{25} + 1812 q^{28} + 1520 q^{34} + 80 q^{37} - 688 q^{40} - 1824 q^{46} - 7904 q^{49} - 5236 q^{52} - 11584 q^{58} - 1648 q^{61} + 5056 q^{64} + 26688 q^{70} + 80 q^{73} - 8388 q^{76} - 38464 q^{82} - 16832 q^{85} - 29520 q^{88} - 4512 q^{94} + 14864 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(55\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.988828 3.87585i −0.247207 0.968963i
\(3\) 0 0
\(4\) −14.0444 + 7.66510i −0.877777 + 0.479069i
\(5\) 20.7568 0.830274 0.415137 0.909759i \(-0.363734\pi\)
0.415137 + 0.909759i \(0.363734\pi\)
\(6\) 0 0
\(7\) 5.38785i 0.109956i −0.998488 0.0549780i \(-0.982491\pi\)
0.998488 0.0549780i \(-0.0175089\pi\)
\(8\) 43.5963 + 46.8547i 0.681192 + 0.732104i
\(9\) 0 0
\(10\) −20.5250 80.4504i −0.205250 0.804504i
\(11\) 115.619i 0.955527i 0.878488 + 0.477764i \(0.158552\pi\)
−0.878488 + 0.477764i \(0.841448\pi\)
\(12\) 0 0
\(13\) 207.234 1.22624 0.613119 0.789991i \(-0.289915\pi\)
0.613119 + 0.789991i \(0.289915\pi\)
\(14\) −20.8825 + 5.32765i −0.106543 + 0.0271819i
\(15\) 0 0
\(16\) 138.493 215.304i 0.540986 0.841031i
\(17\) 383.437 1.32677 0.663385 0.748278i \(-0.269119\pi\)
0.663385 + 0.748278i \(0.269119\pi\)
\(18\) 0 0
\(19\) 618.390i 1.71299i −0.516155 0.856495i \(-0.672637\pi\)
0.516155 0.856495i \(-0.327363\pi\)
\(20\) −291.518 + 159.103i −0.728796 + 0.397758i
\(21\) 0 0
\(22\) 448.121 114.327i 0.925870 0.236213i
\(23\) 82.9899i 0.156881i −0.996919 0.0784403i \(-0.975006\pi\)
0.996919 0.0784403i \(-0.0249940\pi\)
\(24\) 0 0
\(25\) −194.153 −0.310645
\(26\) −204.919 803.209i −0.303135 1.18818i
\(27\) 0 0
\(28\) 41.2984 + 75.6693i 0.0526765 + 0.0965170i
\(29\) 201.949 0.240130 0.120065 0.992766i \(-0.461690\pi\)
0.120065 + 0.992766i \(0.461690\pi\)
\(30\) 0 0
\(31\) 196.140i 0.204100i −0.994779 0.102050i \(-0.967460\pi\)
0.994779 0.102050i \(-0.0325402\pi\)
\(32\) −971.432 323.878i −0.948664 0.316287i
\(33\) 0 0
\(34\) −379.153 1486.14i −0.327987 1.28559i
\(35\) 111.835i 0.0912937i
\(36\) 0 0
\(37\) 340.480 0.248707 0.124354 0.992238i \(-0.460314\pi\)
0.124354 + 0.992238i \(0.460314\pi\)
\(38\) −2396.79 + 611.481i −1.65982 + 0.423463i
\(39\) 0 0
\(40\) 904.922 + 972.556i 0.565576 + 0.607847i
\(41\) 2797.22 1.66402 0.832011 0.554758i \(-0.187189\pi\)
0.832011 + 0.554758i \(0.187189\pi\)
\(42\) 0 0
\(43\) 254.774i 0.137790i −0.997624 0.0688951i \(-0.978053\pi\)
0.997624 0.0688951i \(-0.0219474\pi\)
\(44\) −886.230 1623.80i −0.457763 0.838740i
\(45\) 0 0
\(46\) −321.656 + 82.0627i −0.152011 + 0.0387820i
\(47\) 2257.41i 1.02191i 0.859607 + 0.510956i \(0.170709\pi\)
−0.859607 + 0.510956i \(0.829291\pi\)
\(48\) 0 0
\(49\) 2371.97 0.987910
\(50\) 191.984 + 752.509i 0.0767937 + 0.301004i
\(51\) 0 0
\(52\) −2910.49 + 1588.47i −1.07636 + 0.587452i
\(53\) 4111.23 1.46359 0.731796 0.681524i \(-0.238683\pi\)
0.731796 + 0.681524i \(0.238683\pi\)
\(54\) 0 0
\(55\) 2399.88i 0.793349i
\(56\) 252.446 234.890i 0.0804993 0.0749012i
\(57\) 0 0
\(58\) −199.693 782.726i −0.0593618 0.232677i
\(59\) 1598.57i 0.459226i −0.973282 0.229613i \(-0.926254\pi\)
0.973282 0.229613i \(-0.0737461\pi\)
\(60\) 0 0
\(61\) −6081.09 −1.63426 −0.817131 0.576452i \(-0.804437\pi\)
−0.817131 + 0.576452i \(0.804437\pi\)
\(62\) −760.210 + 193.949i −0.197765 + 0.0504549i
\(63\) 0 0
\(64\) −294.723 + 4085.38i −0.0719539 + 0.997408i
\(65\) 4301.53 1.01811
\(66\) 0 0
\(67\) 7383.67i 1.64484i 0.568884 + 0.822418i \(0.307375\pi\)
−0.568884 + 0.822418i \(0.692625\pi\)
\(68\) −5385.15 + 2939.08i −1.16461 + 0.635614i
\(69\) 0 0
\(70\) −433.455 + 110.585i −0.0884602 + 0.0225684i
\(71\) 9347.39i 1.85427i 0.374723 + 0.927137i \(0.377738\pi\)
−0.374723 + 0.927137i \(0.622262\pi\)
\(72\) 0 0
\(73\) −5320.95 −0.998490 −0.499245 0.866461i \(-0.666389\pi\)
−0.499245 + 0.866461i \(0.666389\pi\)
\(74\) −336.676 1319.65i −0.0614822 0.240988i
\(75\) 0 0
\(76\) 4740.02 + 8684.94i 0.820640 + 1.50362i
\(77\) 622.936 0.105066
\(78\) 0 0
\(79\) 5971.42i 0.956806i −0.878141 0.478403i \(-0.841216\pi\)
0.878141 0.478403i \(-0.158784\pi\)
\(80\) 2874.67 4469.03i 0.449167 0.698286i
\(81\) 0 0
\(82\) −2765.97 10841.6i −0.411358 1.61238i
\(83\) 10091.0i 1.46480i −0.680876 0.732399i \(-0.738401\pi\)
0.680876 0.732399i \(-0.261599\pi\)
\(84\) 0 0
\(85\) 7958.94 1.10158
\(86\) −987.466 + 251.928i −0.133514 + 0.0340627i
\(87\) 0 0
\(88\) −5417.28 + 5040.55i −0.699546 + 0.650898i
\(89\) −13611.5 −1.71841 −0.859206 0.511631i \(-0.829042\pi\)
−0.859206 + 0.511631i \(0.829042\pi\)
\(90\) 0 0
\(91\) 1116.55i 0.134832i
\(92\) 636.126 + 1165.55i 0.0751566 + 0.137706i
\(93\) 0 0
\(94\) 8749.37 2232.19i 0.990195 0.252624i
\(95\) 12835.8i 1.42225i
\(96\) 0 0
\(97\) 2179.54 0.231645 0.115822 0.993270i \(-0.463050\pi\)
0.115822 + 0.993270i \(0.463050\pi\)
\(98\) −2345.47 9193.41i −0.244218 0.957248i
\(99\) 0 0
\(100\) 2726.77 1488.20i 0.272677 0.148820i
\(101\) −3242.55 −0.317866 −0.158933 0.987289i \(-0.550805\pi\)
−0.158933 + 0.987289i \(0.550805\pi\)
\(102\) 0 0
\(103\) 3261.34i 0.307412i −0.988117 0.153706i \(-0.950879\pi\)
0.988117 0.153706i \(-0.0491209\pi\)
\(104\) 9034.65 + 9709.89i 0.835304 + 0.897734i
\(105\) 0 0
\(106\) −4065.30 15934.5i −0.361810 1.41817i
\(107\) 15927.7i 1.39119i 0.718434 + 0.695595i \(0.244859\pi\)
−0.718434 + 0.695595i \(0.755141\pi\)
\(108\) 0 0
\(109\) 7377.06 0.620913 0.310456 0.950588i \(-0.399518\pi\)
0.310456 + 0.950588i \(0.399518\pi\)
\(110\) 9301.58 2373.07i 0.768726 0.196122i
\(111\) 0 0
\(112\) −1160.03 746.177i −0.0924765 0.0594847i
\(113\) −7121.69 −0.557733 −0.278867 0.960330i \(-0.589959\pi\)
−0.278867 + 0.960330i \(0.589959\pi\)
\(114\) 0 0
\(115\) 1722.61i 0.130254i
\(116\) −2836.27 + 1547.96i −0.210781 + 0.115039i
\(117\) 0 0
\(118\) −6195.80 + 1580.71i −0.444973 + 0.113524i
\(119\) 2065.90i 0.145886i
\(120\) 0 0
\(121\) 1273.29 0.0869676
\(122\) 6013.15 + 23569.4i 0.404001 + 1.58354i
\(123\) 0 0
\(124\) 1503.43 + 2754.68i 0.0977779 + 0.179154i
\(125\) −17003.0 −1.08819
\(126\) 0 0
\(127\) 16366.5i 1.01472i −0.861733 0.507362i \(-0.830621\pi\)
0.861733 0.507362i \(-0.169379\pi\)
\(128\) 16125.8 2897.44i 0.984239 0.176846i
\(129\) 0 0
\(130\) −4253.47 16672.1i −0.251685 0.986514i
\(131\) 17497.0i 1.01958i −0.860299 0.509790i \(-0.829723\pi\)
0.860299 0.509790i \(-0.170277\pi\)
\(132\) 0 0
\(133\) −3331.79 −0.188354
\(134\) 28618.0 7301.18i 1.59378 0.406615i
\(135\) 0 0
\(136\) 16716.4 + 17965.8i 0.903786 + 0.971334i
\(137\) −8028.61 −0.427759 −0.213880 0.976860i \(-0.568610\pi\)
−0.213880 + 0.976860i \(0.568610\pi\)
\(138\) 0 0
\(139\) 15001.1i 0.776417i −0.921572 0.388208i \(-0.873094\pi\)
0.921572 0.388208i \(-0.126906\pi\)
\(140\) 857.224 + 1570.66i 0.0437359 + 0.0801355i
\(141\) 0 0
\(142\) 36229.1 9242.96i 1.79672 0.458389i
\(143\) 23960.2i 1.17170i
\(144\) 0 0
\(145\) 4191.83 0.199374
\(146\) 5261.50 + 20623.2i 0.246834 + 0.967499i
\(147\) 0 0
\(148\) −4781.85 + 2609.81i −0.218310 + 0.119148i
\(149\) −1457.01 −0.0656279 −0.0328140 0.999461i \(-0.510447\pi\)
−0.0328140 + 0.999461i \(0.510447\pi\)
\(150\) 0 0
\(151\) 15413.9i 0.676019i 0.941143 + 0.338009i \(0.109754\pi\)
−0.941143 + 0.338009i \(0.890246\pi\)
\(152\) 28974.5 26959.5i 1.25409 1.16688i
\(153\) 0 0
\(154\) −615.977 2414.41i −0.0259731 0.101805i
\(155\) 4071.25i 0.169459i
\(156\) 0 0
\(157\) −23328.9 −0.946443 −0.473222 0.880943i \(-0.656909\pi\)
−0.473222 + 0.880943i \(0.656909\pi\)
\(158\) −23144.3 + 5904.71i −0.927109 + 0.236529i
\(159\) 0 0
\(160\) −20163.9 6722.68i −0.787651 0.262605i
\(161\) −447.137 −0.0172500
\(162\) 0 0
\(163\) 31514.9i 1.18615i 0.805146 + 0.593077i \(0.202087\pi\)
−0.805146 + 0.593077i \(0.797913\pi\)
\(164\) −39285.4 + 21441.0i −1.46064 + 0.797181i
\(165\) 0 0
\(166\) −39111.2 + 9978.26i −1.41933 + 0.362108i
\(167\) 45156.1i 1.61914i −0.587025 0.809569i \(-0.699701\pi\)
0.587025 0.809569i \(-0.300299\pi\)
\(168\) 0 0
\(169\) 14385.0 0.503659
\(170\) −7870.02 30847.6i −0.272319 1.06739i
\(171\) 0 0
\(172\) 1952.87 + 3578.16i 0.0660109 + 0.120949i
\(173\) −39788.6 −1.32943 −0.664716 0.747096i \(-0.731447\pi\)
−0.664716 + 0.747096i \(0.731447\pi\)
\(174\) 0 0
\(175\) 1046.07i 0.0341573i
\(176\) 24893.2 + 16012.3i 0.803628 + 0.516927i
\(177\) 0 0
\(178\) 13459.5 + 52756.3i 0.424803 + 1.66508i
\(179\) 22082.3i 0.689188i 0.938752 + 0.344594i \(0.111983\pi\)
−0.938752 + 0.344594i \(0.888017\pi\)
\(180\) 0 0
\(181\) −9867.01 −0.301181 −0.150591 0.988596i \(-0.548118\pi\)
−0.150591 + 0.988596i \(0.548118\pi\)
\(182\) −4327.57 + 1104.07i −0.130647 + 0.0333315i
\(183\) 0 0
\(184\) 3888.46 3618.05i 0.114853 0.106866i
\(185\) 7067.30 0.206495
\(186\) 0 0
\(187\) 44332.5i 1.26777i
\(188\) −17303.2 31704.0i −0.489566 0.897012i
\(189\) 0 0
\(190\) −49749.7 + 12692.4i −1.37811 + 0.351590i
\(191\) 6985.96i 0.191496i 0.995406 + 0.0957479i \(0.0305243\pi\)
−0.995406 + 0.0957479i \(0.969476\pi\)
\(192\) 0 0
\(193\) −39647.8 −1.06440 −0.532199 0.846619i \(-0.678634\pi\)
−0.532199 + 0.846619i \(0.678634\pi\)
\(194\) −2155.19 8447.59i −0.0572642 0.224455i
\(195\) 0 0
\(196\) −33313.0 + 18181.4i −0.867165 + 0.473277i
\(197\) 25520.9 0.657602 0.328801 0.944399i \(-0.393355\pi\)
0.328801 + 0.944399i \(0.393355\pi\)
\(198\) 0 0
\(199\) 20635.7i 0.521091i −0.965462 0.260546i \(-0.916098\pi\)
0.965462 0.260546i \(-0.0839024\pi\)
\(200\) −8464.36 9096.99i −0.211609 0.227425i
\(201\) 0 0
\(202\) 3206.32 + 12567.6i 0.0785786 + 0.308000i
\(203\) 1088.07i 0.0264038i
\(204\) 0 0
\(205\) 58061.5 1.38159
\(206\) −12640.5 + 3224.90i −0.297871 + 0.0759944i
\(207\) 0 0
\(208\) 28700.4 44618.4i 0.663378 1.03130i
\(209\) 71497.5 1.63681
\(210\) 0 0
\(211\) 37403.1i 0.840123i 0.907496 + 0.420062i \(0.137992\pi\)
−0.907496 + 0.420062i \(0.862008\pi\)
\(212\) −57739.9 + 31513.0i −1.28471 + 0.701161i
\(213\) 0 0
\(214\) 61733.5 15749.8i 1.34801 0.343912i
\(215\) 5288.31i 0.114404i
\(216\) 0 0
\(217\) −1056.77 −0.0224420
\(218\) −7294.65 28592.4i −0.153494 0.601641i
\(219\) 0 0
\(220\) −18395.3 33705.0i −0.380069 0.696384i
\(221\) 79461.2 1.62694
\(222\) 0 0
\(223\) 63893.4i 1.28483i −0.766356 0.642416i \(-0.777932\pi\)
0.766356 0.642416i \(-0.222068\pi\)
\(224\) −1745.00 + 5233.92i −0.0347777 + 0.104311i
\(225\) 0 0
\(226\) 7042.13 + 27602.6i 0.137876 + 0.540423i
\(227\) 52128.0i 1.01162i 0.862644 + 0.505812i \(0.168807\pi\)
−0.862644 + 0.505812i \(0.831193\pi\)
\(228\) 0 0
\(229\) −30234.6 −0.576545 −0.288272 0.957548i \(-0.593081\pi\)
−0.288272 + 0.957548i \(0.593081\pi\)
\(230\) −6676.57 + 1703.36i −0.126211 + 0.0321997i
\(231\) 0 0
\(232\) 8804.25 + 9462.28i 0.163575 + 0.175800i
\(233\) 5080.01 0.0935735 0.0467868 0.998905i \(-0.485102\pi\)
0.0467868 + 0.998905i \(0.485102\pi\)
\(234\) 0 0
\(235\) 46856.6i 0.848468i
\(236\) 12253.2 + 22451.0i 0.220001 + 0.403098i
\(237\) 0 0
\(238\) −8007.11 + 2042.82i −0.141359 + 0.0360641i
\(239\) 40740.6i 0.713234i 0.934251 + 0.356617i \(0.116070\pi\)
−0.934251 + 0.356617i \(0.883930\pi\)
\(240\) 0 0
\(241\) −34834.0 −0.599749 −0.299875 0.953979i \(-0.596945\pi\)
−0.299875 + 0.953979i \(0.596945\pi\)
\(242\) −1259.07 4935.09i −0.0214990 0.0842684i
\(243\) 0 0
\(244\) 85405.5 46612.2i 1.43452 0.782924i
\(245\) 49234.6 0.820236
\(246\) 0 0
\(247\) 128151.i 2.10053i
\(248\) 9190.08 8550.98i 0.149422 0.139031i
\(249\) 0 0
\(250\) 16813.1 + 65901.2i 0.269009 + 1.05442i
\(251\) 71105.6i 1.12864i 0.825555 + 0.564321i \(0.190862\pi\)
−0.825555 + 0.564321i \(0.809138\pi\)
\(252\) 0 0
\(253\) 9595.19 0.149904
\(254\) −63434.1 + 16183.6i −0.983230 + 0.250847i
\(255\) 0 0
\(256\) −27175.6 59636.0i −0.414667 0.909973i
\(257\) −4341.02 −0.0657242 −0.0328621 0.999460i \(-0.510462\pi\)
−0.0328621 + 0.999460i \(0.510462\pi\)
\(258\) 0 0
\(259\) 1834.46i 0.0273469i
\(260\) −60412.6 + 32971.6i −0.893677 + 0.487746i
\(261\) 0 0
\(262\) −67815.8 + 17301.5i −0.987934 + 0.252047i
\(263\) 112382.i 1.62475i −0.583135 0.812375i \(-0.698174\pi\)
0.583135 0.812375i \(-0.301826\pi\)
\(264\) 0 0
\(265\) 85336.1 1.21518
\(266\) 3294.57 + 12913.5i 0.0465624 + 0.182508i
\(267\) 0 0
\(268\) −56596.5 103699.i −0.787989 1.44380i
\(269\) 67178.9 0.928385 0.464193 0.885734i \(-0.346345\pi\)
0.464193 + 0.885734i \(0.346345\pi\)
\(270\) 0 0
\(271\) 30451.7i 0.414641i −0.978273 0.207321i \(-0.933526\pi\)
0.978273 0.207321i \(-0.0664744\pi\)
\(272\) 53103.1 82555.4i 0.717765 1.11586i
\(273\) 0 0
\(274\) 7938.92 + 31117.7i 0.105745 + 0.414483i
\(275\) 22447.8i 0.296830i
\(276\) 0 0
\(277\) 16480.0 0.214782 0.107391 0.994217i \(-0.465750\pi\)
0.107391 + 0.994217i \(0.465750\pi\)
\(278\) −58142.2 + 14833.6i −0.752319 + 0.191936i
\(279\) 0 0
\(280\) 5239.98 4875.58i 0.0668365 0.0621885i
\(281\) 22525.2 0.285270 0.142635 0.989775i \(-0.454442\pi\)
0.142635 + 0.989775i \(0.454442\pi\)
\(282\) 0 0
\(283\) 69282.8i 0.865073i 0.901616 + 0.432537i \(0.142381\pi\)
−0.901616 + 0.432537i \(0.857619\pi\)
\(284\) −71648.7 131279.i −0.888324 1.62764i
\(285\) 0 0
\(286\) 92866.0 23692.5i 1.13534 0.289653i
\(287\) 15071.0i 0.182969i
\(288\) 0 0
\(289\) 63502.6 0.760319
\(290\) −4145.00 16246.9i −0.0492866 0.193186i
\(291\) 0 0
\(292\) 74729.8 40785.6i 0.876452 0.478345i
\(293\) −92504.8 −1.07753 −0.538764 0.842456i \(-0.681109\pi\)
−0.538764 + 0.842456i \(0.681109\pi\)
\(294\) 0 0
\(295\) 33181.2i 0.381283i
\(296\) 14843.7 + 15953.1i 0.169417 + 0.182080i
\(297\) 0 0
\(298\) 1440.73 + 5647.13i 0.0162237 + 0.0635910i
\(299\) 17198.3i 0.192373i
\(300\) 0 0
\(301\) −1372.68 −0.0151509
\(302\) 59742.0 15241.7i 0.655037 0.167117i
\(303\) 0 0
\(304\) −133142. 85642.3i −1.44068 0.926705i
\(305\) −126224. −1.35689
\(306\) 0 0
\(307\) 79741.9i 0.846077i −0.906112 0.423038i \(-0.860964\pi\)
0.906112 0.423038i \(-0.139036\pi\)
\(308\) −8748.79 + 4774.87i −0.0922246 + 0.0503338i
\(309\) 0 0
\(310\) −15779.6 + 4025.76i −0.164199 + 0.0418914i
\(311\) 115631.i 1.19551i 0.801679 + 0.597755i \(0.203940\pi\)
−0.801679 + 0.597755i \(0.796060\pi\)
\(312\) 0 0
\(313\) −25857.6 −0.263936 −0.131968 0.991254i \(-0.542130\pi\)
−0.131968 + 0.991254i \(0.542130\pi\)
\(314\) 23068.2 + 90419.2i 0.233967 + 0.917068i
\(315\) 0 0
\(316\) 45771.6 + 83865.3i 0.458376 + 0.839862i
\(317\) −78300.9 −0.779198 −0.389599 0.920985i \(-0.627386\pi\)
−0.389599 + 0.920985i \(0.627386\pi\)
\(318\) 0 0
\(319\) 23349.1i 0.229451i
\(320\) −6117.53 + 84799.7i −0.0597415 + 0.828122i
\(321\) 0 0
\(322\) 442.141 + 1733.04i 0.00426432 + 0.0167146i
\(323\) 237113.i 2.27274i
\(324\) 0 0
\(325\) −40235.2 −0.380925
\(326\) 122147. 31162.8i 1.14934 0.293226i
\(327\) 0 0
\(328\) 121949. + 131063.i 1.13352 + 1.21824i
\(329\) 12162.6 0.112366
\(330\) 0 0
\(331\) 159345.i 1.45439i 0.686429 + 0.727197i \(0.259177\pi\)
−0.686429 + 0.727197i \(0.740823\pi\)
\(332\) 77348.5 + 141722.i 0.701739 + 1.28577i
\(333\) 0 0
\(334\) −175018. + 44651.6i −1.56888 + 0.400262i
\(335\) 153262.i 1.36566i
\(336\) 0 0
\(337\) 32887.8 0.289584 0.144792 0.989462i \(-0.453749\pi\)
0.144792 + 0.989462i \(0.453749\pi\)
\(338\) −14224.3 55754.2i −0.124508 0.488027i
\(339\) 0 0
\(340\) −111779. + 61006.0i −0.966944 + 0.527734i
\(341\) 22677.5 0.195023
\(342\) 0 0
\(343\) 25716.0i 0.218583i
\(344\) 11937.4 11107.2i 0.100877 0.0938616i
\(345\) 0 0
\(346\) 39344.0 + 154215.i 0.328645 + 1.28817i
\(347\) 68679.4i 0.570385i 0.958470 + 0.285192i \(0.0920575\pi\)
−0.958470 + 0.285192i \(0.907943\pi\)
\(348\) 0 0
\(349\) −228554. −1.87645 −0.938226 0.346024i \(-0.887532\pi\)
−0.938226 + 0.346024i \(0.887532\pi\)
\(350\) 4054.40 1034.38i 0.0330972 0.00844393i
\(351\) 0 0
\(352\) 37446.4 112316.i 0.302221 0.906474i
\(353\) −46025.5 −0.369359 −0.184680 0.982799i \(-0.559125\pi\)
−0.184680 + 0.982799i \(0.559125\pi\)
\(354\) 0 0
\(355\) 194022.i 1.53956i
\(356\) 191166. 104334.i 1.50838 0.823237i
\(357\) 0 0
\(358\) 85587.6 21835.6i 0.667797 0.170372i
\(359\) 131445.i 1.01990i −0.860205 0.509948i \(-0.829665\pi\)
0.860205 0.509948i \(-0.170335\pi\)
\(360\) 0 0
\(361\) −252085. −1.93434
\(362\) 9756.77 + 38243.0i 0.0744542 + 0.291834i
\(363\) 0 0
\(364\) 8558.44 + 15681.3i 0.0645939 + 0.118353i
\(365\) −110446. −0.829020
\(366\) 0 0
\(367\) 120649.i 0.895756i −0.894095 0.447878i \(-0.852180\pi\)
0.894095 0.447878i \(-0.147820\pi\)
\(368\) −17868.1 11493.5i −0.131942 0.0848703i
\(369\) 0 0
\(370\) −6988.34 27391.8i −0.0510470 0.200086i
\(371\) 22150.7i 0.160931i
\(372\) 0 0
\(373\) 183241. 1.31706 0.658530 0.752555i \(-0.271179\pi\)
0.658530 + 0.752555i \(0.271179\pi\)
\(374\) 171826. 43837.2i 1.22842 0.313400i
\(375\) 0 0
\(376\) −105770. + 98414.6i −0.748147 + 0.696119i
\(377\) 41850.8 0.294457
\(378\) 0 0
\(379\) 210539.i 1.46573i 0.680374 + 0.732865i \(0.261817\pi\)
−0.680374 + 0.732865i \(0.738183\pi\)
\(380\) 98387.8 + 180272.i 0.681356 + 1.24842i
\(381\) 0 0
\(382\) 27076.5 6907.91i 0.185552 0.0473391i
\(383\) 115027.i 0.784153i 0.919933 + 0.392076i \(0.128243\pi\)
−0.919933 + 0.392076i \(0.871757\pi\)
\(384\) 0 0
\(385\) 12930.2 0.0872336
\(386\) 39204.8 + 153669.i 0.263127 + 1.03136i
\(387\) 0 0
\(388\) −30610.5 + 16706.4i −0.203332 + 0.110974i
\(389\) −227952. −1.50641 −0.753206 0.657785i \(-0.771493\pi\)
−0.753206 + 0.657785i \(0.771493\pi\)
\(390\) 0 0
\(391\) 31821.4i 0.208145i
\(392\) 103409. + 111138.i 0.672957 + 0.723253i
\(393\) 0 0
\(394\) −25235.8 98915.2i −0.162564 0.637192i
\(395\) 123948.i 0.794411i
\(396\) 0 0
\(397\) 253211. 1.60657 0.803287 0.595592i \(-0.203082\pi\)
0.803287 + 0.595592i \(0.203082\pi\)
\(398\) −79981.0 + 20405.2i −0.504918 + 0.128817i
\(399\) 0 0
\(400\) −26888.8 + 41802.0i −0.168055 + 0.261262i
\(401\) −113054. −0.703071 −0.351535 0.936175i \(-0.614340\pi\)
−0.351535 + 0.936175i \(0.614340\pi\)
\(402\) 0 0
\(403\) 40646.9i 0.250275i
\(404\) 45539.7 24854.4i 0.279015 0.152279i
\(405\) 0 0
\(406\) −4217.21 + 1075.92i −0.0255843 + 0.00652719i
\(407\) 39365.9i 0.237647i
\(408\) 0 0
\(409\) 77602.4 0.463905 0.231952 0.972727i \(-0.425489\pi\)
0.231952 + 0.972727i \(0.425489\pi\)
\(410\) −57412.9 225038.i −0.341540 1.33871i
\(411\) 0 0
\(412\) 24998.5 + 45803.6i 0.147272 + 0.269839i
\(413\) −8612.83 −0.0504947
\(414\) 0 0
\(415\) 209457.i 1.21618i
\(416\) −201314. 67118.5i −1.16329 0.387843i
\(417\) 0 0
\(418\) −70698.7 277114.i −0.404631 1.58601i
\(419\) 46895.2i 0.267116i 0.991041 + 0.133558i \(0.0426403\pi\)
−0.991041 + 0.133558i \(0.957360\pi\)
\(420\) 0 0
\(421\) 40148.2 0.226518 0.113259 0.993566i \(-0.463871\pi\)
0.113259 + 0.993566i \(0.463871\pi\)
\(422\) 144969. 36985.3i 0.814048 0.207684i
\(423\) 0 0
\(424\) 179234. + 192630.i 0.996987 + 1.07150i
\(425\) −74445.5 −0.412155
\(426\) 0 0
\(427\) 32764.0i 0.179697i
\(428\) −122088. 223696.i −0.666475 1.22115i
\(429\) 0 0
\(430\) −20496.7 + 5229.22i −0.110853 + 0.0282814i
\(431\) 257425.i 1.38579i −0.721040 0.692893i \(-0.756336\pi\)
0.721040 0.692893i \(-0.243664\pi\)
\(432\) 0 0
\(433\) −29760.6 −0.158732 −0.0793661 0.996846i \(-0.525290\pi\)
−0.0793661 + 0.996846i \(0.525290\pi\)
\(434\) 1044.97 + 4095.89i 0.00554783 + 0.0217455i
\(435\) 0 0
\(436\) −103607. + 56545.9i −0.545023 + 0.297460i
\(437\) −51320.1 −0.268735
\(438\) 0 0
\(439\) 249198.i 1.29305i −0.762892 0.646526i \(-0.776221\pi\)
0.762892 0.646526i \(-0.223779\pi\)
\(440\) −112446. + 104626.i −0.580815 + 0.540424i
\(441\) 0 0
\(442\) −78573.4 307980.i −0.402190 1.57644i
\(443\) 53833.8i 0.274314i 0.990549 + 0.137157i \(0.0437964\pi\)
−0.990549 + 0.137157i \(0.956204\pi\)
\(444\) 0 0
\(445\) −282533. −1.42675
\(446\) −247641. + 63179.6i −1.24495 + 0.317619i
\(447\) 0 0
\(448\) 22011.4 + 1587.92i 0.109671 + 0.00791177i
\(449\) 127961. 0.634726 0.317363 0.948304i \(-0.397203\pi\)
0.317363 + 0.948304i \(0.397203\pi\)
\(450\) 0 0
\(451\) 323412.i 1.59002i
\(452\) 100020. 54588.5i 0.489566 0.267192i
\(453\) 0 0
\(454\) 202040. 51545.6i 0.980227 0.250081i
\(455\) 23176.0i 0.111948i
\(456\) 0 0
\(457\) 138874. 0.664951 0.332475 0.943112i \(-0.392116\pi\)
0.332475 + 0.943112i \(0.392116\pi\)
\(458\) 29896.8 + 117185.i 0.142526 + 0.558650i
\(459\) 0 0
\(460\) 13204.0 + 24193.1i 0.0624006 + 0.114334i
\(461\) 242960. 1.14323 0.571613 0.820523i \(-0.306318\pi\)
0.571613 + 0.820523i \(0.306318\pi\)
\(462\) 0 0
\(463\) 277264.i 1.29340i −0.762746 0.646699i \(-0.776149\pi\)
0.762746 0.646699i \(-0.223851\pi\)
\(464\) 27968.5 43480.5i 0.129907 0.201957i
\(465\) 0 0
\(466\) −5023.26 19689.4i −0.0231320 0.0906692i
\(467\) 123580.i 0.566649i −0.959024 0.283325i \(-0.908563\pi\)
0.959024 0.283325i \(-0.0914374\pi\)
\(468\) 0 0
\(469\) 39782.1 0.180860
\(470\) 181609. 46333.1i 0.822133 0.209747i
\(471\) 0 0
\(472\) 74900.3 69691.6i 0.336202 0.312821i
\(473\) 29456.7 0.131662
\(474\) 0 0
\(475\) 120062.i 0.532132i
\(476\) 15835.3 + 29014.4i 0.0698896 + 0.128056i
\(477\) 0 0
\(478\) 157905. 40285.5i 0.691097 0.176316i
\(479\) 56214.6i 0.245007i 0.992468 + 0.122503i \(0.0390922\pi\)
−0.992468 + 0.122503i \(0.960908\pi\)
\(480\) 0 0
\(481\) 70559.1 0.304974
\(482\) 34444.9 + 135011.i 0.148262 + 0.581134i
\(483\) 0 0
\(484\) −17882.7 + 9759.92i −0.0763382 + 0.0416635i
\(485\) 45240.5 0.192328
\(486\) 0 0
\(487\) 271980.i 1.14678i −0.819283 0.573389i \(-0.805628\pi\)
0.819283 0.573389i \(-0.194372\pi\)
\(488\) −265113. 284928.i −1.11325 1.19645i
\(489\) 0 0
\(490\) −48684.6 190826.i −0.202768 0.794778i
\(491\) 335208.i 1.39044i 0.718798 + 0.695219i \(0.244693\pi\)
−0.718798 + 0.695219i \(0.755307\pi\)
\(492\) 0 0
\(493\) 77434.8 0.318597
\(494\) −496696. + 126720.i −2.03534 + 0.519267i
\(495\) 0 0
\(496\) −42229.7 27163.9i −0.171654 0.110415i
\(497\) 50362.3 0.203889
\(498\) 0 0
\(499\) 287692.i 1.15538i −0.816255 0.577692i \(-0.803953\pi\)
0.816255 0.577692i \(-0.196047\pi\)
\(500\) 238798. 130330.i 0.955193 0.521320i
\(501\) 0 0
\(502\) 275595. 70311.2i 1.09361 0.279008i
\(503\) 338788.i 1.33903i 0.742797 + 0.669517i \(0.233499\pi\)
−0.742797 + 0.669517i \(0.766501\pi\)
\(504\) 0 0
\(505\) −67305.0 −0.263915
\(506\) −9487.99 37189.5i −0.0370572 0.145251i
\(507\) 0 0
\(508\) 125451. + 229858.i 0.486123 + 0.890702i
\(509\) 444920. 1.71730 0.858651 0.512561i \(-0.171303\pi\)
0.858651 + 0.512561i \(0.171303\pi\)
\(510\) 0 0
\(511\) 28668.5i 0.109790i
\(512\) −204268. + 164298.i −0.779221 + 0.626749i
\(513\) 0 0
\(514\) 4292.52 + 16825.1i 0.0162475 + 0.0636843i
\(515\) 67695.1i 0.255236i
\(516\) 0 0
\(517\) −260999. −0.976466
\(518\) −7110.08 + 1813.96i −0.0264981 + 0.00676034i
\(519\) 0 0
\(520\) 187531. + 201547.i 0.693531 + 0.745365i
\(521\) 275911. 1.01647 0.508233 0.861220i \(-0.330299\pi\)
0.508233 + 0.861220i \(0.330299\pi\)
\(522\) 0 0
\(523\) 85712.3i 0.313357i −0.987650 0.156679i \(-0.949921\pi\)
0.987650 0.156679i \(-0.0500787\pi\)
\(524\) 134116. + 245736.i 0.488448 + 0.894964i
\(525\) 0 0
\(526\) −435577. + 111127.i −1.57432 + 0.401650i
\(527\) 75207.3i 0.270794i
\(528\) 0 0
\(529\) 272954. 0.975388
\(530\) −84382.7 330750.i −0.300401 1.17747i
\(531\) 0 0
\(532\) 46793.1 25538.5i 0.165333 0.0902344i
\(533\) 579680. 2.04049
\(534\) 0 0
\(535\) 330610.i 1.15507i
\(536\) −345959. + 321901.i −1.20419 + 1.12045i
\(537\) 0 0
\(538\) −66428.4 260375.i −0.229503 0.899571i
\(539\) 274244.i 0.943975i
\(540\) 0 0
\(541\) 250046. 0.854331 0.427165 0.904174i \(-0.359512\pi\)
0.427165 + 0.904174i \(0.359512\pi\)
\(542\) −118026. + 30111.5i −0.401772 + 0.102502i
\(543\) 0 0
\(544\) −372482. 124187.i −1.25866 0.419640i
\(545\) 153125. 0.515528
\(546\) 0 0
\(547\) 435437.i 1.45529i 0.685951 + 0.727647i \(0.259386\pi\)
−0.685951 + 0.727647i \(0.740614\pi\)
\(548\) 112757. 61540.1i 0.375477 0.204926i
\(549\) 0 0
\(550\) −87004.2 + 22197.0i −0.287617 + 0.0733784i
\(551\) 124883.i 0.411341i
\(552\) 0 0
\(553\) −32173.1 −0.105207
\(554\) −16295.9 63874.2i −0.0530957 0.208116i
\(555\) 0 0
\(556\) 114985. + 210683.i 0.371957 + 0.681521i
\(557\) −466836. −1.50471 −0.752356 0.658756i \(-0.771083\pi\)
−0.752356 + 0.658756i \(0.771083\pi\)
\(558\) 0 0
\(559\) 52797.9i 0.168964i
\(560\) −24078.5 15488.3i −0.0767808 0.0493886i
\(561\) 0 0
\(562\) −22273.6 87304.4i −0.0705208 0.276416i
\(563\) 75191.2i 0.237219i −0.992941 0.118610i \(-0.962156\pi\)
0.992941 0.118610i \(-0.0378437\pi\)
\(564\) 0 0
\(565\) −147824. −0.463071
\(566\) 268530. 68508.8i 0.838224 0.213852i
\(567\) 0 0
\(568\) −437969. + 407512.i −1.35752 + 1.26312i
\(569\) −66289.8 −0.204749 −0.102375 0.994746i \(-0.532644\pi\)
−0.102375 + 0.994746i \(0.532644\pi\)
\(570\) 0 0
\(571\) 377844.i 1.15888i 0.815014 + 0.579442i \(0.196729\pi\)
−0.815014 + 0.579442i \(0.803271\pi\)
\(572\) −183657. 336507.i −0.561327 1.02850i
\(573\) 0 0
\(574\) −58413.0 + 14902.6i −0.177291 + 0.0452313i
\(575\) 16112.8i 0.0487342i
\(576\) 0 0
\(577\) −415556. −1.24818 −0.624091 0.781352i \(-0.714530\pi\)
−0.624091 + 0.781352i \(0.714530\pi\)
\(578\) −62793.2 246127.i −0.187956 0.736721i
\(579\) 0 0
\(580\) −58872.0 + 32130.8i −0.175006 + 0.0955137i
\(581\) −54368.7 −0.161063
\(582\) 0 0
\(583\) 475335.i 1.39850i
\(584\) −231974. 249311.i −0.680163 0.730999i
\(585\) 0 0
\(586\) 91471.3 + 358535.i 0.266373 + 1.04409i
\(587\) 342605.i 0.994301i −0.867664 0.497151i \(-0.834380\pi\)
0.867664 0.497151i \(-0.165620\pi\)
\(588\) 0 0
\(589\) −121291. −0.349621
\(590\) −128605. + 32810.5i −0.369449 + 0.0942559i
\(591\) 0 0
\(592\) 47154.0 73306.8i 0.134547 0.209171i
\(593\) 125568. 0.357082 0.178541 0.983932i \(-0.442862\pi\)
0.178541 + 0.983932i \(0.442862\pi\)
\(594\) 0 0
\(595\) 42881.5i 0.121126i
\(596\) 20462.8 11168.1i 0.0576067 0.0314403i
\(597\) 0 0
\(598\) −66658.2 + 17006.2i −0.186402 + 0.0475559i
\(599\) 317723.i 0.885512i −0.896642 0.442756i \(-0.854001\pi\)
0.896642 0.442756i \(-0.145999\pi\)
\(600\) 0 0
\(601\) −168972. −0.467806 −0.233903 0.972260i \(-0.575150\pi\)
−0.233903 + 0.972260i \(0.575150\pi\)
\(602\) 1357.35 + 5320.32i 0.00374540 + 0.0146806i
\(603\) 0 0
\(604\) −118149. 216480.i −0.323859 0.593394i
\(605\) 26429.5 0.0722070
\(606\) 0 0
\(607\) 644373.i 1.74888i −0.485135 0.874440i \(-0.661229\pi\)
0.485135 0.874440i \(-0.338771\pi\)
\(608\) −200283. + 600723.i −0.541796 + 1.62505i
\(609\) 0 0
\(610\) 124814. + 489227.i 0.335432 + 1.31477i
\(611\) 467812.i 1.25311i
\(612\) 0 0
\(613\) 228892. 0.609129 0.304564 0.952492i \(-0.401489\pi\)
0.304564 + 0.952492i \(0.401489\pi\)
\(614\) −309068. + 78851.0i −0.819817 + 0.209156i
\(615\) 0 0
\(616\) 27157.7 + 29187.5i 0.0715702 + 0.0769193i
\(617\) −329605. −0.865813 −0.432906 0.901439i \(-0.642512\pi\)
−0.432906 + 0.901439i \(0.642512\pi\)
\(618\) 0 0
\(619\) 223932.i 0.584434i −0.956352 0.292217i \(-0.905607\pi\)
0.956352 0.292217i \(-0.0943930\pi\)
\(620\) 31206.5 + 57178.4i 0.0811824 + 0.148747i
\(621\) 0 0
\(622\) 448168. 114339.i 1.15840 0.295538i
\(623\) 73336.9i 0.188950i
\(624\) 0 0
\(625\) −231584. −0.592854
\(626\) 25568.7 + 100220.i 0.0652468 + 0.255744i
\(627\) 0 0
\(628\) 327641. 178818.i 0.830766 0.453411i
\(629\) 130553. 0.329977
\(630\) 0 0
\(631\) 681430.i 1.71144i 0.517437 + 0.855721i \(0.326886\pi\)
−0.517437 + 0.855721i \(0.673114\pi\)
\(632\) 279789. 260332.i 0.700482 0.651769i
\(633\) 0 0
\(634\) 77426.1 + 303482.i 0.192623 + 0.755014i
\(635\) 339717.i 0.842499i
\(636\) 0 0
\(637\) 491554. 1.21141
\(638\) 90497.8 23088.3i 0.222329 0.0567219i
\(639\) 0 0
\(640\) 334720. 60141.7i 0.817188 0.146830i
\(641\) −346761. −0.843945 −0.421973 0.906609i \(-0.638662\pi\)
−0.421973 + 0.906609i \(0.638662\pi\)
\(642\) 0 0
\(643\) 124932.i 0.302169i −0.988521 0.151085i \(-0.951723\pi\)
0.988521 0.151085i \(-0.0482766\pi\)
\(644\) 6279.78 3427.35i 0.0151416 0.00826392i
\(645\) 0 0
\(646\) −919015. + 234464.i −2.20221 + 0.561838i
\(647\) 43450.9i 0.103798i 0.998652 + 0.0518992i \(0.0165274\pi\)
−0.998652 + 0.0518992i \(0.983473\pi\)
\(648\) 0 0
\(649\) 184824. 0.438803
\(650\) 39785.7 + 155946.i 0.0941673 + 0.369102i
\(651\) 0 0
\(652\) −241565. 442609.i −0.568249 1.04118i
\(653\) −344790. −0.808589 −0.404294 0.914629i \(-0.632483\pi\)
−0.404294 + 0.914629i \(0.632483\pi\)
\(654\) 0 0
\(655\) 363183.i 0.846530i
\(656\) 387394. 602253.i 0.900214 1.39950i
\(657\) 0 0
\(658\) −12026.7 47140.3i −0.0277775 0.108878i
\(659\) 670185.i 1.54321i 0.636105 + 0.771603i \(0.280545\pi\)
−0.636105 + 0.771603i \(0.719455\pi\)
\(660\) 0 0
\(661\) −619584. −1.41807 −0.709034 0.705174i \(-0.750869\pi\)
−0.709034 + 0.705174i \(0.750869\pi\)
\(662\) 617597. 157565.i 1.40925 0.359536i
\(663\) 0 0
\(664\) 472810. 439930.i 1.07239 0.997809i
\(665\) −69157.4 −0.156385
\(666\) 0 0
\(667\) 16759.8i 0.0376718i
\(668\) 346126. + 634193.i 0.775678 + 1.42124i
\(669\) 0 0
\(670\) 594019. 151549.i 1.32328 0.337602i
\(671\) 703088.i 1.56158i
\(672\) 0 0
\(673\) −233848. −0.516302 −0.258151 0.966105i \(-0.583113\pi\)
−0.258151 + 0.966105i \(0.583113\pi\)
\(674\) −32520.3 127468.i −0.0715872 0.280596i
\(675\) 0 0
\(676\) −202029. + 110263.i −0.442101 + 0.241287i
\(677\) −703621. −1.53519 −0.767594 0.640936i \(-0.778546\pi\)
−0.767594 + 0.640936i \(0.778546\pi\)
\(678\) 0 0
\(679\) 11743.0i 0.0254707i
\(680\) 346980. + 372913.i 0.750390 + 0.806474i
\(681\) 0 0
\(682\) −22424.1 87894.5i −0.0482111 0.188970i
\(683\) 8903.15i 0.0190855i −0.999954 0.00954273i \(-0.996962\pi\)
0.999954 0.00954273i \(-0.00303759\pi\)
\(684\) 0 0
\(685\) −166649. −0.355157
\(686\) −99671.5 + 25428.7i −0.211799 + 0.0540352i
\(687\) 0 0
\(688\) −54853.9 35284.3i −0.115886 0.0745426i
\(689\) 851987. 1.79471
\(690\) 0 0
\(691\) 186507.i 0.390606i −0.980743 0.195303i \(-0.937431\pi\)
0.980743 0.195303i \(-0.0625690\pi\)
\(692\) 558808. 304983.i 1.16694 0.636889i
\(693\) 0 0
\(694\) 266191. 67912.1i 0.552681 0.141003i
\(695\) 311376.i 0.644638i
\(696\) 0 0
\(697\) 1.07256e6 2.20778
\(698\) 226000. + 885840.i 0.463872 + 1.81821i
\(699\) 0 0
\(700\) −8018.21 14691.4i −0.0163637 0.0299825i
\(701\) −498328. −1.01410 −0.507048 0.861918i \(-0.669264\pi\)
−0.507048 + 0.861918i \(0.669264\pi\)
\(702\) 0 0
\(703\) 210549.i 0.426033i
\(704\) −472347. 34075.5i −0.953051 0.0687539i
\(705\) 0 0
\(706\) 45511.3 + 178388.i 0.0913082 + 0.357895i
\(707\) 17470.3i 0.0349512i
\(708\) 0 0
\(709\) 84448.4 0.167996 0.0839980 0.996466i \(-0.473231\pi\)
0.0839980 + 0.996466i \(0.473231\pi\)
\(710\) 752002. 191855.i 1.49177 0.380589i
\(711\) 0 0
\(712\) −593413. 637764.i −1.17057 1.25806i
\(713\) −16277.6 −0.0320193
\(714\) 0 0
\(715\) 497338.i 0.972835i
\(716\) −169263. 310133.i −0.330168 0.604954i
\(717\) 0 0
\(718\) −509462. + 129977.i −0.988241 + 0.252125i
\(719\) 563153.i 1.08935i 0.838646 + 0.544677i \(0.183348\pi\)
−0.838646 + 0.544677i \(0.816652\pi\)
\(720\) 0 0
\(721\) −17571.6 −0.0338018
\(722\) 249268. + 977043.i 0.478182 + 1.87430i
\(723\) 0 0
\(724\) 138577. 75631.6i 0.264370 0.144287i
\(725\) −39209.1 −0.0745953
\(726\) 0 0
\(727\) 829063.i 1.56862i −0.620367 0.784312i \(-0.713016\pi\)
0.620367 0.784312i \(-0.286984\pi\)
\(728\) 52315.4 48677.3i 0.0987113 0.0918467i
\(729\) 0 0
\(730\) 109212. + 428073.i 0.204939 + 0.803289i
\(731\) 97689.7i 0.182816i
\(732\) 0 0
\(733\) 345658. 0.643338 0.321669 0.946852i \(-0.395756\pi\)
0.321669 + 0.946852i \(0.395756\pi\)
\(734\) −467616. + 119301.i −0.867954 + 0.221437i
\(735\) 0 0
\(736\) −26878.6 + 80619.0i −0.0496193 + 0.148827i
\(737\) −853691. −1.57169
\(738\) 0 0
\(739\) 662859.i 1.21376i −0.794794 0.606879i \(-0.792421\pi\)
0.794794 0.606879i \(-0.207579\pi\)
\(740\) −99256.2 + 54171.5i −0.181257 + 0.0989253i
\(741\) 0 0
\(742\) −85852.7 + 21903.2i −0.155936 + 0.0397832i
\(743\) 346708.i 0.628039i 0.949417 + 0.314019i \(0.101676\pi\)
−0.949417 + 0.314019i \(0.898324\pi\)
\(744\) 0 0
\(745\) −30242.8 −0.0544891
\(746\) −181194. 710216.i −0.325586 1.27618i
\(747\) 0 0
\(748\) −339813. 622625.i −0.607347 1.11282i
\(749\) 85816.2 0.152970
\(750\) 0 0
\(751\) 218129.i 0.386752i 0.981125 + 0.193376i \(0.0619438\pi\)
−0.981125 + 0.193376i \(0.938056\pi\)
\(752\) 486028. + 312634.i 0.859461 + 0.552841i
\(753\) 0 0
\(754\) −41383.3 162208.i −0.0727917 0.285318i
\(755\) 319944.i 0.561281i
\(756\) 0 0
\(757\) 454177. 0.792563 0.396281 0.918129i \(-0.370300\pi\)
0.396281 + 0.918129i \(0.370300\pi\)
\(758\) 816017. 208187.i 1.42024 0.362339i
\(759\) 0 0
\(760\) 601418. 559594.i 1.04124 0.968827i
\(761\) −779754. −1.34644 −0.673222 0.739441i \(-0.735090\pi\)
−0.673222 + 0.739441i \(0.735090\pi\)
\(762\) 0 0
\(763\) 39746.5i 0.0682731i
\(764\) −53548.1 98113.9i −0.0917396 0.168091i
\(765\) 0 0
\(766\) 445826. 113742.i 0.759815 0.193848i
\(767\) 331278.i 0.563120i
\(768\) 0 0
\(769\) 1.04561e6 1.76814 0.884072 0.467350i \(-0.154791\pi\)
0.884072 + 0.467350i \(0.154791\pi\)
\(770\) −12785.7 50115.5i −0.0215647 0.0845261i
\(771\) 0 0
\(772\) 556831. 303904.i 0.934305 0.509920i
\(773\) 240697. 0.402820 0.201410 0.979507i \(-0.435448\pi\)
0.201410 + 0.979507i \(0.435448\pi\)
\(774\) 0 0
\(775\) 38081.2i 0.0634027i
\(776\) 95020.1 + 102122.i 0.157795 + 0.169588i
\(777\) 0 0
\(778\) 225405. + 883506.i 0.372395 + 1.45966i
\(779\) 1.72977e6i 2.85046i
\(780\) 0 0
\(781\) −1.08073e6 −1.77181
\(782\) −123335. + 31465.8i −0.201684 + 0.0514548i
\(783\) 0 0
\(784\) 328500. 510695.i 0.534446 0.830863i
\(785\) −484234. −0.785807
\(786\) 0 0
\(787\) 453844.i 0.732753i −0.930467 0.366376i \(-0.880598\pi\)
0.930467 0.366376i \(-0.119402\pi\)
\(788\) −358427. + 195620.i −0.577229 + 0.315037i
\(789\) 0 0
\(790\) −480404. + 122563.i −0.769754 + 0.196384i
\(791\) 38370.6i 0.0613261i
\(792\) 0 0
\(793\) −1.26021e6 −2.00399
\(794\) −250382. 981406.i −0.397156 1.55671i
\(795\) 0 0
\(796\) 158175. + 289817.i 0.249638 + 0.457402i
\(797\) −595530. −0.937534 −0.468767 0.883322i \(-0.655302\pi\)
−0.468767 + 0.883322i \(0.655302\pi\)
\(798\) 0 0
\(799\) 865572.i 1.35584i
\(800\) 188607. + 62881.9i 0.294698 + 0.0982530i
\(801\) 0 0
\(802\) 111791. + 438182.i 0.173804 + 0.681249i
\(803\) 615202.i 0.954084i
\(804\) 0 0
\(805\) −9281.15 −0.0143222
\(806\) −157541. + 40192.8i −0.242507 + 0.0618697i
\(807\) 0 0
\(808\) −141363. 151928.i −0.216528 0.232711i
\(809\) 238455. 0.364342 0.182171 0.983267i \(-0.441688\pi\)
0.182171 + 0.983267i \(0.441688\pi\)
\(810\) 0 0
\(811\) 325863.i 0.495443i 0.968831 + 0.247721i \(0.0796817\pi\)
−0.968831 + 0.247721i \(0.920318\pi\)
\(812\) 8340.18 + 15281.4i 0.0126492 + 0.0231766i
\(813\) 0 0
\(814\) 152576. 38926.1i 0.230271 0.0587479i
\(815\) 654151.i 0.984833i
\(816\) 0 0
\(817\) −157550. −0.236033
\(818\) −76735.4 300775.i −0.114680 0.449506i
\(819\) 0 0
\(820\) −815442. + 445047.i −1.21273 + 0.661879i
\(821\) 376943. 0.559228 0.279614 0.960113i \(-0.409794\pi\)
0.279614 + 0.960113i \(0.409794\pi\)
\(822\) 0 0
\(823\) 1.10349e6i 1.62918i 0.580040 + 0.814588i \(0.303037\pi\)
−0.580040 + 0.814588i \(0.696963\pi\)
\(824\) 152809. 142182.i 0.225058 0.209407i
\(825\) 0 0
\(826\) 8516.61 + 33382.0i 0.0124826 + 0.0489275i
\(827\) 251807.i 0.368177i 0.982910 + 0.184089i \(0.0589334\pi\)
−0.982910 + 0.184089i \(0.941067\pi\)
\(828\) 0 0
\(829\) 595280. 0.866189 0.433094 0.901349i \(-0.357422\pi\)
0.433094 + 0.901349i \(0.357422\pi\)
\(830\) −811825. + 207117.i −1.17844 + 0.300649i
\(831\) 0 0
\(832\) −61076.7 + 846631.i −0.0882326 + 1.22306i
\(833\) 909501. 1.31073
\(834\) 0 0
\(835\) 937299.i 1.34433i
\(836\) −1.00414e6 + 548035.i −1.43675 + 0.784144i
\(837\) 0 0
\(838\) 181759. 46371.3i 0.258826 0.0660330i
\(839\) 937926.i 1.33243i 0.745760 + 0.666215i \(0.232087\pi\)
−0.745760 + 0.666215i \(0.767913\pi\)
\(840\) 0 0
\(841\) −666497. −0.942338
\(842\) −39699.7 155609.i −0.0559967 0.219487i
\(843\) 0 0
\(844\) −286699. 525306.i −0.402477 0.737441i
\(845\) 298588. 0.418175
\(846\) 0 0
\(847\) 6860.31i 0.00956262i
\(848\) 569374. 885164.i 0.791783 1.23093i
\(849\) 0 0
\(850\) 73613.7 + 288539.i 0.101888 + 0.399363i
\(851\) 28256.4i 0.0390174i
\(852\) 0 0
\(853\) −17614.3 −0.0242085 −0.0121042 0.999927i \(-0.503853\pi\)
−0.0121042 + 0.999927i \(0.503853\pi\)
\(854\) 126988. 32397.9i 0.174120 0.0444224i
\(855\) 0 0
\(856\) −746289. + 694390.i −1.01850 + 0.947668i
\(857\) 203626. 0.277251 0.138625 0.990345i \(-0.455732\pi\)
0.138625 + 0.990345i \(0.455732\pi\)
\(858\) 0 0
\(859\) 110138.i 0.149262i −0.997211 0.0746310i \(-0.976222\pi\)
0.997211 0.0746310i \(-0.0237779\pi\)
\(860\) 40535.4 + 74271.3i 0.0548072 + 0.100421i
\(861\) 0 0
\(862\) −997741. + 254549.i −1.34278 + 0.342576i
\(863\) 366628.i 0.492271i 0.969235 + 0.246136i \(0.0791609\pi\)
−0.969235 + 0.246136i \(0.920839\pi\)
\(864\) 0 0
\(865\) −825885. −1.10379
\(866\) 29428.1 + 115347.i 0.0392397 + 0.153806i
\(867\) 0 0
\(868\) 14841.8 8100.27i 0.0196991 0.0107513i
\(869\) 690409. 0.914254
\(870\) 0 0
\(871\) 1.53015e6i 2.01696i
\(872\) 321613. + 345650.i 0.422961 + 0.454573i
\(873\) 0 0
\(874\) 50746.7 + 198909.i 0.0664332 + 0.260394i
\(875\) 91609.8i 0.119654i
\(876\) 0 0
\(877\) −556086. −0.723007 −0.361503 0.932371i \(-0.617736\pi\)
−0.361503 + 0.932371i \(0.617736\pi\)
\(878\) −965855. + 246414.i −1.25292 + 0.319651i
\(879\) 0 0
\(880\) 516704. + 332366.i 0.667232 + 0.429191i
\(881\) −246117. −0.317096 −0.158548 0.987351i \(-0.550681\pi\)
−0.158548 + 0.987351i \(0.550681\pi\)
\(882\) 0 0
\(883\) 317186.i 0.406810i −0.979095 0.203405i \(-0.934799\pi\)
0.979095 0.203405i \(-0.0652009\pi\)
\(884\) −1.11599e6 + 609078.i −1.42809 + 0.779414i
\(885\) 0 0
\(886\) 208652. 53232.3i 0.265800 0.0678122i
\(887\) 84967.8i 0.107996i −0.998541 0.0539979i \(-0.982804\pi\)
0.998541 0.0539979i \(-0.0171964\pi\)
\(888\) 0 0
\(889\) −88180.2 −0.111575
\(890\) 279376. + 1.09505e6i 0.352703 + 1.38247i
\(891\) 0 0
\(892\) 489749. + 897347.i 0.615523 + 1.12780i
\(893\) 1.39596e6 1.75053
\(894\) 0 0
\(895\) 458358.i 0.572215i
\(896\) −15610.9 86883.2i −0.0194452 0.108223i
\(897\) 0 0
\(898\) −126532. 495959.i −0.156909 0.615026i
\(899\) 39610.4i 0.0490105i
\(900\) 0 0
\(901\) 1.57640e6 1.94185
\(902\) 1.25349e6 319798.i 1.54067 0.393064i
\(903\) 0 0
\(904\) −310480. 333685.i −0.379924 0.408319i
\(905\) −204808. −0.250063
\(906\) 0 0
\(907\) 384478.i 0.467366i 0.972313 + 0.233683i \(0.0750778\pi\)
−0.972313 + 0.233683i \(0.924922\pi\)
\(908\) −399566. 732109.i −0.484638 0.887981i
\(909\) 0 0
\(910\) −89826.7 + 22917.1i −0.108473 + 0.0276743i
\(911\) 1.24116e6i 1.49552i −0.663972 0.747758i \(-0.731130\pi\)
0.663972 0.747758i \(-0.268870\pi\)
\(912\) 0 0
\(913\) 1.16671e6 1.39965
\(914\) −137323. 538256.i −0.164380 0.644312i
\(915\) 0 0
\(916\) 424628. 231751.i 0.506078 0.276205i
\(917\) −94271.2 −0.112109
\(918\) 0 0
\(919\) 698308.i 0.826829i 0.910543 + 0.413415i \(0.135664\pi\)
−0.910543 + 0.413415i \(0.864336\pi\)
\(920\) 80712.3 75099.4i 0.0953595 0.0887280i
\(921\) 0 0
\(922\) −240245. 941675.i −0.282614 1.10774i
\(923\) 1.93710e6i 2.27378i
\(924\) 0 0
\(925\) −66105.3 −0.0772597
\(926\) −1.07463e6 + 274167.i −1.25325 + 0.319737i
\(927\) 0 0
\(928\) −196180. 65406.9i −0.227803 0.0759500i
\(929\) 1.40938e6 1.63304 0.816519 0.577318i \(-0.195901\pi\)
0.816519 + 0.577318i \(0.195901\pi\)
\(930\) 0 0
\(931\) 1.46680e6i 1.69228i
\(932\) −71345.9 + 38938.8i −0.0821367 + 0.0448281i
\(933\) 0 0
\(934\) −478978. + 122199.i −0.549062 + 0.140080i
\(935\) 920203.i 1.05259i
\(936\) 0 0
\(937\) −98815.5 −0.112550 −0.0562751 0.998415i \(-0.517922\pi\)
−0.0562751 + 0.998415i \(0.517922\pi\)
\(938\) −39337.6 154189.i −0.0447098 0.175246i
\(939\) 0 0
\(940\) −359161. 658075.i −0.406474 0.744766i
\(941\) 1.30446e6 1.47316 0.736581 0.676350i \(-0.236439\pi\)
0.736581 + 0.676350i \(0.236439\pi\)
\(942\) 0 0
\(943\) 232141.i 0.261053i
\(944\) −344178. 221389.i −0.386224 0.248435i
\(945\) 0 0
\(946\) −29127.6 114170.i −0.0325478 0.127576i
\(947\) 146295.i 0.163128i 0.996668 + 0.0815641i \(0.0259915\pi\)
−0.996668 + 0.0815641i \(0.974008\pi\)
\(948\) 0 0
\(949\) −1.10268e6 −1.22439
\(950\) 465344. 118721.i 0.515616 0.131547i
\(951\) 0 0
\(952\) 96797.0 90065.5i 0.106804 0.0993767i
\(953\) 1.65688e6 1.82433 0.912167 0.409818i \(-0.134408\pi\)
0.912167 + 0.409818i \(0.134408\pi\)
\(954\) 0 0
\(955\) 145006.i 0.158994i
\(956\) −312281. 572179.i −0.341688 0.626061i
\(957\) 0 0
\(958\) 217879. 55586.5i 0.237402 0.0605674i
\(959\) 43256.9i 0.0470347i
\(960\) 0 0
\(961\) 885050. 0.958343
\(962\) −69770.8 273477.i −0.0753918 0.295509i
\(963\) 0 0
\(964\) 489224. 267006.i 0.526446 0.287321i
\(965\) −822963. −0.883742
\(966\) 0 0
\(967\) 532588.i 0.569559i 0.958593 + 0.284780i \(0.0919204\pi\)
−0.958593 + 0.284780i \(0.908080\pi\)
\(968\) 55510.9 + 59659.7i 0.0592417 + 0.0636694i
\(969\) 0 0
\(970\) −44735.0 175345.i −0.0475449 0.186359i
\(971\) 1.26243e6i 1.33896i −0.742828 0.669482i \(-0.766516\pi\)
0.742828 0.669482i \(-0.233484\pi\)
\(972\) 0 0
\(973\) −80823.9 −0.0853717
\(974\) −1.05416e6 + 268942.i −1.11119 + 0.283492i
\(975\) 0 0
\(976\) −842186. + 1.30928e6i −0.884114 + 1.37447i
\(977\) −995130. −1.04254 −0.521268 0.853393i \(-0.674541\pi\)
−0.521268 + 0.853393i \(0.674541\pi\)
\(978\) 0 0
\(979\) 1.57375e6i 1.64199i
\(980\) −691473. + 377388.i −0.719984 + 0.392949i
\(981\) 0 0
\(982\) 1.29922e6 331463.i 1.34728 0.343726i
\(983\) 706269.i 0.730909i 0.930829 + 0.365454i \(0.119086\pi\)
−0.930829 + 0.365454i \(0.880914\pi\)
\(984\) 0 0
\(985\) 529733. 0.545990
\(986\) −76569.7 300126.i −0.0787595 0.308709i
\(987\) 0 0
\(988\) 982294. + 1.79982e6i 1.00630 + 1.84380i
\(989\) −21143.7 −0.0216166
\(990\) 0 0
\(991\) 6734.63i 0.00685751i −0.999994 0.00342875i \(-0.998909\pi\)
0.999994 0.00342875i \(-0.00109141\pi\)
\(992\) −63525.4 + 190537.i −0.0645541 + 0.193622i
\(993\) 0 0
\(994\) −49799.7 195197.i −0.0504027 0.197560i
\(995\) 428333.i 0.432648i
\(996\) 0 0
\(997\) −525528. −0.528696 −0.264348 0.964427i \(-0.585157\pi\)
−0.264348 + 0.964427i \(0.585157\pi\)
\(998\) −1.11505e6 + 284478.i −1.11952 + 0.285619i
\(999\) 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 108.5.d.b.55.7 16
3.2 odd 2 inner 108.5.d.b.55.10 yes 16
4.3 odd 2 inner 108.5.d.b.55.8 yes 16
12.11 even 2 inner 108.5.d.b.55.9 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.5.d.b.55.7 16 1.1 even 1 trivial
108.5.d.b.55.8 yes 16 4.3 odd 2 inner
108.5.d.b.55.9 yes 16 12.11 even 2 inner
108.5.d.b.55.10 yes 16 3.2 odd 2 inner