Properties

Label 1089.3.c.k.604.6
Level $1089$
Weight $3$
Character 1089.604
Analytic conductor $29.673$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1089,3,Mod(604,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1089.604");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1089.c (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: \(29.6731007888\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.523388583936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 28x^{6} + 262x^{4} + 948x^{2} + 1089 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 11 \)
Twist minimal: no (minimal twist has level 121)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 604.6
Root \(2.16205i\) of defining polynomial
Character \(\chi\) \(=\) 1089.604
Dual form 1089.3.c.k.604.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.16205i q^{2} -0.674453 q^{4} -7.64086 q^{5} -6.05557i q^{7} +7.18999i q^{8} +O(q^{10})\) \(q+2.16205i q^{2} -0.674453 q^{4} -7.64086 q^{5} -6.05557i q^{7} +7.18999i q^{8} -16.5199i q^{10} +3.21180i q^{13} +13.0924 q^{14} -18.2429 q^{16} +6.22852i q^{17} -19.4519i q^{19} +5.15340 q^{20} -3.47096 q^{23} +33.3827 q^{25} -6.94407 q^{26} +4.08419i q^{28} -36.1250i q^{29} -3.28901 q^{31} -10.6821i q^{32} -13.4664 q^{34} +46.2697i q^{35} +63.2118 q^{37} +42.0560 q^{38} -54.9377i q^{40} -31.9231i q^{41} +43.9407i q^{43} -7.50439i q^{46} +34.7799 q^{47} +12.3301 q^{49} +72.1750i q^{50} -2.16621i q^{52} -16.5478 q^{53} +43.5395 q^{56} +78.1040 q^{58} +63.7608 q^{59} +110.544i q^{61} -7.11100i q^{62} -49.8765 q^{64} -24.5409i q^{65} +96.1439 q^{67} -4.20084i q^{68} -100.037 q^{70} +44.8264 q^{71} +63.2763i q^{73} +136.667i q^{74} +13.1194i q^{76} -14.1046i q^{79} +139.392 q^{80} +69.0193 q^{82} +88.6120i q^{83} -47.5912i q^{85} -95.0018 q^{86} -51.3977 q^{89} +19.4493 q^{91} +2.34100 q^{92} +75.1958i q^{94} +148.630i q^{95} +31.5024 q^{97} +26.6583i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{4} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{4} + 4 q^{5} + 4 q^{14} - 24 q^{16} - 52 q^{20} + 12 q^{23} - 16 q^{25} - 168 q^{26} - 116 q^{31} - 180 q^{34} - 4 q^{37} + 132 q^{38} + 244 q^{47} + 88 q^{49} - 268 q^{53} + 12 q^{56} + 88 q^{58} + 56 q^{59} - 40 q^{64} + 284 q^{67} - 188 q^{70} - 272 q^{71} + 356 q^{80} - 180 q^{82} - 336 q^{86} + 24 q^{89} + 140 q^{91} + 156 q^{92} + 152 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(848\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.16205i 1.08102i 0.841336 + 0.540512i \(0.181769\pi\)
−0.841336 + 0.540512i \(0.818231\pi\)
\(3\) 0 0
\(4\) −0.674453 −0.168613
\(5\) −7.64086 −1.52817 −0.764086 0.645115i \(-0.776810\pi\)
−0.764086 + 0.645115i \(0.776810\pi\)
\(6\) 0 0
\(7\) − 6.05557i − 0.865081i −0.901615 0.432540i \(-0.857617\pi\)
0.901615 0.432540i \(-0.142383\pi\)
\(8\) 7.18999i 0.898749i
\(9\) 0 0
\(10\) − 16.5199i − 1.65199i
\(11\) 0 0
\(12\) 0 0
\(13\) 3.21180i 0.247062i 0.992341 + 0.123531i \(0.0394218\pi\)
−0.992341 + 0.123531i \(0.960578\pi\)
\(14\) 13.0924 0.935173
\(15\) 0 0
\(16\) −18.2429 −1.14018
\(17\) 6.22852i 0.366383i 0.983077 + 0.183192i \(0.0586429\pi\)
−0.983077 + 0.183192i \(0.941357\pi\)
\(18\) 0 0
\(19\) − 19.4519i − 1.02379i −0.859049 0.511893i \(-0.828944\pi\)
0.859049 0.511893i \(-0.171056\pi\)
\(20\) 5.15340 0.257670
\(21\) 0 0
\(22\) 0 0
\(23\) −3.47096 −0.150912 −0.0754558 0.997149i \(-0.524041\pi\)
−0.0754558 + 0.997149i \(0.524041\pi\)
\(24\) 0 0
\(25\) 33.3827 1.33531
\(26\) −6.94407 −0.267080
\(27\) 0 0
\(28\) 4.08419i 0.145864i
\(29\) − 36.1250i − 1.24569i −0.782345 0.622845i \(-0.785977\pi\)
0.782345 0.622845i \(-0.214023\pi\)
\(30\) 0 0
\(31\) −3.28901 −0.106097 −0.0530486 0.998592i \(-0.516894\pi\)
−0.0530486 + 0.998592i \(0.516894\pi\)
\(32\) − 10.6821i − 0.333816i
\(33\) 0 0
\(34\) −13.4664 −0.396069
\(35\) 46.2697i 1.32199i
\(36\) 0 0
\(37\) 63.2118 1.70843 0.854214 0.519922i \(-0.174039\pi\)
0.854214 + 0.519922i \(0.174039\pi\)
\(38\) 42.0560 1.10674
\(39\) 0 0
\(40\) − 54.9377i − 1.37344i
\(41\) − 31.9231i − 0.778612i −0.921108 0.389306i \(-0.872715\pi\)
0.921108 0.389306i \(-0.127285\pi\)
\(42\) 0 0
\(43\) 43.9407i 1.02188i 0.859618 + 0.510938i \(0.170702\pi\)
−0.859618 + 0.510938i \(0.829298\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) − 7.50439i − 0.163139i
\(47\) 34.7799 0.739998 0.369999 0.929032i \(-0.379358\pi\)
0.369999 + 0.929032i \(0.379358\pi\)
\(48\) 0 0
\(49\) 12.3301 0.251635
\(50\) 72.1750i 1.44350i
\(51\) 0 0
\(52\) − 2.16621i − 0.0416578i
\(53\) −16.5478 −0.312223 −0.156111 0.987739i \(-0.549896\pi\)
−0.156111 + 0.987739i \(0.549896\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 43.5395 0.777491
\(57\) 0 0
\(58\) 78.1040 1.34662
\(59\) 63.7608 1.08069 0.540346 0.841443i \(-0.318293\pi\)
0.540346 + 0.841443i \(0.318293\pi\)
\(60\) 0 0
\(61\) 110.544i 1.81220i 0.423059 + 0.906102i \(0.360956\pi\)
−0.423059 + 0.906102i \(0.639044\pi\)
\(62\) − 7.11100i − 0.114694i
\(63\) 0 0
\(64\) −49.8765 −0.779320
\(65\) − 24.5409i − 0.377553i
\(66\) 0 0
\(67\) 96.1439 1.43498 0.717492 0.696567i \(-0.245290\pi\)
0.717492 + 0.696567i \(0.245290\pi\)
\(68\) − 4.20084i − 0.0617771i
\(69\) 0 0
\(70\) −100.037 −1.42911
\(71\) 44.8264 0.631358 0.315679 0.948866i \(-0.397768\pi\)
0.315679 + 0.948866i \(0.397768\pi\)
\(72\) 0 0
\(73\) 63.2763i 0.866798i 0.901202 + 0.433399i \(0.142686\pi\)
−0.901202 + 0.433399i \(0.857314\pi\)
\(74\) 136.667i 1.84685i
\(75\) 0 0
\(76\) 13.1194i 0.172624i
\(77\) 0 0
\(78\) 0 0
\(79\) − 14.1046i − 0.178540i −0.996007 0.0892699i \(-0.971547\pi\)
0.996007 0.0892699i \(-0.0284534\pi\)
\(80\) 139.392 1.74239
\(81\) 0 0
\(82\) 69.0193 0.841699
\(83\) 88.6120i 1.06761i 0.845606 + 0.533807i \(0.179239\pi\)
−0.845606 + 0.533807i \(0.820761\pi\)
\(84\) 0 0
\(85\) − 47.5912i − 0.559897i
\(86\) −95.0018 −1.10467
\(87\) 0 0
\(88\) 0 0
\(89\) −51.3977 −0.577502 −0.288751 0.957404i \(-0.593240\pi\)
−0.288751 + 0.957404i \(0.593240\pi\)
\(90\) 0 0
\(91\) 19.4493 0.213728
\(92\) 2.34100 0.0254457
\(93\) 0 0
\(94\) 75.1958i 0.799956i
\(95\) 148.630i 1.56452i
\(96\) 0 0
\(97\) 31.5024 0.324767 0.162383 0.986728i \(-0.448082\pi\)
0.162383 + 0.986728i \(0.448082\pi\)
\(98\) 26.6583i 0.272023i
\(99\) 0 0
\(100\) −22.5151 −0.225151
\(101\) 84.1059i 0.832732i 0.909197 + 0.416366i \(0.136696\pi\)
−0.909197 + 0.416366i \(0.863304\pi\)
\(102\) 0 0
\(103\) 76.6144 0.743829 0.371915 0.928267i \(-0.378701\pi\)
0.371915 + 0.928267i \(0.378701\pi\)
\(104\) −23.0928 −0.222046
\(105\) 0 0
\(106\) − 35.7771i − 0.337520i
\(107\) − 128.437i − 1.20034i −0.799872 0.600171i \(-0.795099\pi\)
0.799872 0.600171i \(-0.204901\pi\)
\(108\) 0 0
\(109\) − 150.418i − 1.37998i −0.723819 0.689990i \(-0.757615\pi\)
0.723819 0.689990i \(-0.242385\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 110.471i 0.986350i
\(113\) 102.830 0.910004 0.455002 0.890490i \(-0.349638\pi\)
0.455002 + 0.890490i \(0.349638\pi\)
\(114\) 0 0
\(115\) 26.5212 0.230619
\(116\) 24.3646i 0.210040i
\(117\) 0 0
\(118\) 137.854i 1.16825i
\(119\) 37.7172 0.316951
\(120\) 0 0
\(121\) 0 0
\(122\) −239.002 −1.95904
\(123\) 0 0
\(124\) 2.21828 0.0178894
\(125\) −64.0511 −0.512409
\(126\) 0 0
\(127\) − 231.035i − 1.81917i −0.415517 0.909585i \(-0.636399\pi\)
0.415517 0.909585i \(-0.363601\pi\)
\(128\) − 150.564i − 1.17628i
\(129\) 0 0
\(130\) 53.0586 0.408143
\(131\) − 234.208i − 1.78784i −0.448223 0.893922i \(-0.647943\pi\)
0.448223 0.893922i \(-0.352057\pi\)
\(132\) 0 0
\(133\) −117.793 −0.885658
\(134\) 207.868i 1.55125i
\(135\) 0 0
\(136\) −44.7830 −0.329287
\(137\) 145.175 1.05967 0.529837 0.848100i \(-0.322253\pi\)
0.529837 + 0.848100i \(0.322253\pi\)
\(138\) 0 0
\(139\) 20.6006i 0.148206i 0.997251 + 0.0741028i \(0.0236093\pi\)
−0.997251 + 0.0741028i \(0.976391\pi\)
\(140\) − 31.2067i − 0.222905i
\(141\) 0 0
\(142\) 96.9169i 0.682513i
\(143\) 0 0
\(144\) 0 0
\(145\) 276.026i 1.90363i
\(146\) −136.806 −0.937030
\(147\) 0 0
\(148\) −42.6334 −0.288063
\(149\) 80.6247i 0.541105i 0.962705 + 0.270553i \(0.0872064\pi\)
−0.962705 + 0.270553i \(0.912794\pi\)
\(150\) 0 0
\(151\) 150.438i 0.996276i 0.867098 + 0.498138i \(0.165983\pi\)
−0.867098 + 0.498138i \(0.834017\pi\)
\(152\) 139.859 0.920127
\(153\) 0 0
\(154\) 0 0
\(155\) 25.1309 0.162135
\(156\) 0 0
\(157\) −81.4919 −0.519057 −0.259528 0.965735i \(-0.583567\pi\)
−0.259528 + 0.965735i \(0.583567\pi\)
\(158\) 30.4949 0.193006
\(159\) 0 0
\(160\) 81.6205i 0.510128i
\(161\) 21.0187i 0.130551i
\(162\) 0 0
\(163\) −23.0774 −0.141579 −0.0707895 0.997491i \(-0.522552\pi\)
−0.0707895 + 0.997491i \(0.522552\pi\)
\(164\) 21.5306i 0.131284i
\(165\) 0 0
\(166\) −191.584 −1.15412
\(167\) − 292.437i − 1.75112i −0.483110 0.875560i \(-0.660493\pi\)
0.483110 0.875560i \(-0.339507\pi\)
\(168\) 0 0
\(169\) 158.684 0.938961
\(170\) 102.895 0.605262
\(171\) 0 0
\(172\) − 29.6359i − 0.172302i
\(173\) 212.290i 1.22711i 0.789651 + 0.613556i \(0.210262\pi\)
−0.789651 + 0.613556i \(0.789738\pi\)
\(174\) 0 0
\(175\) − 202.151i − 1.15515i
\(176\) 0 0
\(177\) 0 0
\(178\) − 111.124i − 0.624294i
\(179\) −52.8561 −0.295285 −0.147643 0.989041i \(-0.547169\pi\)
−0.147643 + 0.989041i \(0.547169\pi\)
\(180\) 0 0
\(181\) −26.2830 −0.145210 −0.0726051 0.997361i \(-0.523131\pi\)
−0.0726051 + 0.997361i \(0.523131\pi\)
\(182\) 42.0503i 0.231045i
\(183\) 0 0
\(184\) − 24.9562i − 0.135632i
\(185\) −482.992 −2.61077
\(186\) 0 0
\(187\) 0 0
\(188\) −23.4574 −0.124773
\(189\) 0 0
\(190\) −321.344 −1.69129
\(191\) 115.146 0.602861 0.301430 0.953488i \(-0.402536\pi\)
0.301430 + 0.953488i \(0.402536\pi\)
\(192\) 0 0
\(193\) − 38.6666i − 0.200345i −0.994970 0.100172i \(-0.968061\pi\)
0.994970 0.100172i \(-0.0319395\pi\)
\(194\) 68.1097i 0.351081i
\(195\) 0 0
\(196\) −8.31608 −0.0424290
\(197\) 3.42417i 0.0173815i 0.999962 + 0.00869077i \(0.00276639\pi\)
−0.999962 + 0.00869077i \(0.997234\pi\)
\(198\) 0 0
\(199\) −183.079 −0.919997 −0.459999 0.887920i \(-0.652150\pi\)
−0.459999 + 0.887920i \(0.652150\pi\)
\(200\) 240.021i 1.20011i
\(201\) 0 0
\(202\) −181.841 −0.900203
\(203\) −218.757 −1.07762
\(204\) 0 0
\(205\) 243.920i 1.18985i
\(206\) 165.644i 0.804097i
\(207\) 0 0
\(208\) − 58.5926i − 0.281695i
\(209\) 0 0
\(210\) 0 0
\(211\) − 271.800i − 1.28815i −0.764961 0.644077i \(-0.777242\pi\)
0.764961 0.644077i \(-0.222758\pi\)
\(212\) 11.1607 0.0526449
\(213\) 0 0
\(214\) 277.686 1.29760
\(215\) − 335.744i − 1.56160i
\(216\) 0 0
\(217\) 19.9168i 0.0917826i
\(218\) 325.211 1.49179
\(219\) 0 0
\(220\) 0 0
\(221\) −20.0048 −0.0905193
\(222\) 0 0
\(223\) 44.7473 0.200661 0.100330 0.994954i \(-0.468010\pi\)
0.100330 + 0.994954i \(0.468010\pi\)
\(224\) −64.6862 −0.288778
\(225\) 0 0
\(226\) 222.324i 0.983737i
\(227\) 131.114i 0.577596i 0.957390 + 0.288798i \(0.0932556\pi\)
−0.957390 + 0.288798i \(0.906744\pi\)
\(228\) 0 0
\(229\) −276.449 −1.20720 −0.603601 0.797287i \(-0.706268\pi\)
−0.603601 + 0.797287i \(0.706268\pi\)
\(230\) 57.3400i 0.249304i
\(231\) 0 0
\(232\) 259.739 1.11956
\(233\) − 42.8283i − 0.183812i −0.995768 0.0919061i \(-0.970704\pi\)
0.995768 0.0919061i \(-0.0292960\pi\)
\(234\) 0 0
\(235\) −265.748 −1.13084
\(236\) −43.0037 −0.182219
\(237\) 0 0
\(238\) 81.5464i 0.342632i
\(239\) 13.1360i 0.0549625i 0.999622 + 0.0274812i \(0.00874865\pi\)
−0.999622 + 0.0274812i \(0.991251\pi\)
\(240\) 0 0
\(241\) 80.4946i 0.334002i 0.985957 + 0.167001i \(0.0534084\pi\)
−0.985957 + 0.167001i \(0.946592\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) − 74.5570i − 0.305561i
\(245\) −94.2127 −0.384541
\(246\) 0 0
\(247\) 62.4758 0.252938
\(248\) − 23.6480i − 0.0953547i
\(249\) 0 0
\(250\) − 138.482i − 0.553927i
\(251\) 311.802 1.24224 0.621120 0.783716i \(-0.286678\pi\)
0.621120 + 0.783716i \(0.286678\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 499.508 1.96657
\(255\) 0 0
\(256\) 126.020 0.492267
\(257\) 24.7875 0.0964496 0.0482248 0.998837i \(-0.484644\pi\)
0.0482248 + 0.998837i \(0.484644\pi\)
\(258\) 0 0
\(259\) − 382.783i − 1.47793i
\(260\) 16.5517i 0.0636603i
\(261\) 0 0
\(262\) 506.368 1.93270
\(263\) 189.407i 0.720178i 0.932918 + 0.360089i \(0.117254\pi\)
−0.932918 + 0.360089i \(0.882746\pi\)
\(264\) 0 0
\(265\) 126.439 0.477130
\(266\) − 254.673i − 0.957418i
\(267\) 0 0
\(268\) −64.8445 −0.241957
\(269\) 409.266 1.52144 0.760718 0.649082i \(-0.224847\pi\)
0.760718 + 0.649082i \(0.224847\pi\)
\(270\) 0 0
\(271\) 24.5223i 0.0904882i 0.998976 + 0.0452441i \(0.0144066\pi\)
−0.998976 + 0.0452441i \(0.985593\pi\)
\(272\) − 113.626i − 0.417744i
\(273\) 0 0
\(274\) 313.876i 1.14553i
\(275\) 0 0
\(276\) 0 0
\(277\) − 80.5213i − 0.290691i −0.989381 0.145345i \(-0.953571\pi\)
0.989381 0.145345i \(-0.0464293\pi\)
\(278\) −44.5395 −0.160214
\(279\) 0 0
\(280\) −332.679 −1.18814
\(281\) − 58.2626i − 0.207340i −0.994612 0.103670i \(-0.966941\pi\)
0.994612 0.103670i \(-0.0330586\pi\)
\(282\) 0 0
\(283\) − 124.525i − 0.440016i −0.975498 0.220008i \(-0.929392\pi\)
0.975498 0.220008i \(-0.0706084\pi\)
\(284\) −30.2333 −0.106455
\(285\) 0 0
\(286\) 0 0
\(287\) −193.312 −0.673563
\(288\) 0 0
\(289\) 250.206 0.865763
\(290\) −596.782 −2.05787
\(291\) 0 0
\(292\) − 42.6769i − 0.146154i
\(293\) − 99.0592i − 0.338086i −0.985609 0.169043i \(-0.945932\pi\)
0.985609 0.169043i \(-0.0540677\pi\)
\(294\) 0 0
\(295\) −487.187 −1.65148
\(296\) 454.492i 1.53545i
\(297\) 0 0
\(298\) −174.314 −0.584948
\(299\) − 11.1480i − 0.0372844i
\(300\) 0 0
\(301\) 266.086 0.884005
\(302\) −325.254 −1.07700
\(303\) 0 0
\(304\) 354.860i 1.16730i
\(305\) − 844.654i − 2.76936i
\(306\) 0 0
\(307\) 335.303i 1.09219i 0.837723 + 0.546096i \(0.183886\pi\)
−0.837723 + 0.546096i \(0.816114\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 54.3342i 0.175271i
\(311\) −257.049 −0.826525 −0.413263 0.910612i \(-0.635611\pi\)
−0.413263 + 0.910612i \(0.635611\pi\)
\(312\) 0 0
\(313\) 187.670 0.599585 0.299792 0.954004i \(-0.403083\pi\)
0.299792 + 0.954004i \(0.403083\pi\)
\(314\) − 176.190i − 0.561113i
\(315\) 0 0
\(316\) 9.51291i 0.0301042i
\(317\) −324.893 −1.02490 −0.512449 0.858718i \(-0.671262\pi\)
−0.512449 + 0.858718i \(0.671262\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 381.099 1.19093
\(321\) 0 0
\(322\) −45.4434 −0.141128
\(323\) 121.157 0.375098
\(324\) 0 0
\(325\) 107.219i 0.329903i
\(326\) − 49.8944i − 0.153050i
\(327\) 0 0
\(328\) 229.527 0.699777
\(329\) − 210.612i − 0.640158i
\(330\) 0 0
\(331\) 599.240 1.81039 0.905196 0.424993i \(-0.139724\pi\)
0.905196 + 0.424993i \(0.139724\pi\)
\(332\) − 59.7646i − 0.180014i
\(333\) 0 0
\(334\) 632.263 1.89300
\(335\) −734.622 −2.19290
\(336\) 0 0
\(337\) − 480.354i − 1.42538i −0.701477 0.712692i \(-0.747476\pi\)
0.701477 0.712692i \(-0.252524\pi\)
\(338\) 343.083i 1.01504i
\(339\) 0 0
\(340\) 32.0980i 0.0944060i
\(341\) 0 0
\(342\) 0 0
\(343\) − 371.389i − 1.08277i
\(344\) −315.933 −0.918410
\(345\) 0 0
\(346\) −458.982 −1.32654
\(347\) 21.1483i 0.0609461i 0.999536 + 0.0304731i \(0.00970138\pi\)
−0.999536 + 0.0304731i \(0.990299\pi\)
\(348\) 0 0
\(349\) − 361.648i − 1.03624i −0.855307 0.518121i \(-0.826632\pi\)
0.855307 0.518121i \(-0.173368\pi\)
\(350\) 437.061 1.24874
\(351\) 0 0
\(352\) 0 0
\(353\) −140.448 −0.397871 −0.198935 0.980013i \(-0.563748\pi\)
−0.198935 + 0.980013i \(0.563748\pi\)
\(354\) 0 0
\(355\) −342.512 −0.964823
\(356\) 34.6653 0.0973745
\(357\) 0 0
\(358\) − 114.277i − 0.319211i
\(359\) − 433.079i − 1.20635i −0.797609 0.603174i \(-0.793902\pi\)
0.797609 0.603174i \(-0.206098\pi\)
\(360\) 0 0
\(361\) −17.3783 −0.0481392
\(362\) − 56.8252i − 0.156976i
\(363\) 0 0
\(364\) −13.1176 −0.0360374
\(365\) − 483.485i − 1.32462i
\(366\) 0 0
\(367\) 22.0832 0.0601723 0.0300862 0.999547i \(-0.490422\pi\)
0.0300862 + 0.999547i \(0.490422\pi\)
\(368\) 63.3206 0.172067
\(369\) 0 0
\(370\) − 1044.25i − 2.82231i
\(371\) 100.206i 0.270098i
\(372\) 0 0
\(373\) − 485.424i − 1.30140i −0.759333 0.650702i \(-0.774475\pi\)
0.759333 0.650702i \(-0.225525\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 250.067i 0.665072i
\(377\) 116.026 0.307762
\(378\) 0 0
\(379\) −710.615 −1.87497 −0.937487 0.348021i \(-0.886854\pi\)
−0.937487 + 0.348021i \(0.886854\pi\)
\(380\) − 100.244i − 0.263799i
\(381\) 0 0
\(382\) 248.952i 0.651707i
\(383\) −70.1703 −0.183212 −0.0916061 0.995795i \(-0.529200\pi\)
−0.0916061 + 0.995795i \(0.529200\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 83.5990 0.216578
\(387\) 0 0
\(388\) −21.2469 −0.0547600
\(389\) 503.306 1.29385 0.646923 0.762555i \(-0.276055\pi\)
0.646923 + 0.762555i \(0.276055\pi\)
\(390\) 0 0
\(391\) − 21.6190i − 0.0552915i
\(392\) 88.6534i 0.226157i
\(393\) 0 0
\(394\) −7.40321 −0.0187899
\(395\) 107.772i 0.272839i
\(396\) 0 0
\(397\) −144.659 −0.364380 −0.182190 0.983263i \(-0.558319\pi\)
−0.182190 + 0.983263i \(0.558319\pi\)
\(398\) − 395.827i − 0.994539i
\(399\) 0 0
\(400\) −608.998 −1.52250
\(401\) −598.115 −1.49156 −0.745779 0.666194i \(-0.767922\pi\)
−0.745779 + 0.666194i \(0.767922\pi\)
\(402\) 0 0
\(403\) − 10.5636i − 0.0262125i
\(404\) − 56.7255i − 0.140410i
\(405\) 0 0
\(406\) − 472.964i − 1.16494i
\(407\) 0 0
\(408\) 0 0
\(409\) 209.088i 0.511217i 0.966780 + 0.255609i \(0.0822758\pi\)
−0.966780 + 0.255609i \(0.917724\pi\)
\(410\) −527.367 −1.28626
\(411\) 0 0
\(412\) −51.6728 −0.125419
\(413\) − 386.108i − 0.934886i
\(414\) 0 0
\(415\) − 677.072i − 1.63150i
\(416\) 34.3088 0.0824731
\(417\) 0 0
\(418\) 0 0
\(419\) 170.309 0.406466 0.203233 0.979130i \(-0.434855\pi\)
0.203233 + 0.979130i \(0.434855\pi\)
\(420\) 0 0
\(421\) 301.865 0.717019 0.358510 0.933526i \(-0.383285\pi\)
0.358510 + 0.933526i \(0.383285\pi\)
\(422\) 587.646 1.39253
\(423\) 0 0
\(424\) − 118.979i − 0.280610i
\(425\) 207.925i 0.489235i
\(426\) 0 0
\(427\) 669.409 1.56770
\(428\) 86.6244i 0.202393i
\(429\) 0 0
\(430\) 725.895 1.68813
\(431\) − 416.880i − 0.967238i −0.875279 0.483619i \(-0.839322\pi\)
0.875279 0.483619i \(-0.160678\pi\)
\(432\) 0 0
\(433\) −296.975 −0.685854 −0.342927 0.939362i \(-0.611418\pi\)
−0.342927 + 0.939362i \(0.611418\pi\)
\(434\) −43.0611 −0.0992192
\(435\) 0 0
\(436\) 101.450i 0.232683i
\(437\) 67.5170i 0.154501i
\(438\) 0 0
\(439\) 690.951i 1.57392i 0.617004 + 0.786960i \(0.288346\pi\)
−0.617004 + 0.786960i \(0.711654\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 43.2513i − 0.0978535i
\(443\) 769.034 1.73597 0.867984 0.496592i \(-0.165415\pi\)
0.867984 + 0.496592i \(0.165415\pi\)
\(444\) 0 0
\(445\) 392.722 0.882522
\(446\) 96.7459i 0.216919i
\(447\) 0 0
\(448\) 302.030i 0.674175i
\(449\) 350.143 0.779827 0.389914 0.920851i \(-0.372505\pi\)
0.389914 + 0.920851i \(0.372505\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −69.3543 −0.153439
\(453\) 0 0
\(454\) −283.475 −0.624395
\(455\) −148.609 −0.326613
\(456\) 0 0
\(457\) 424.881i 0.929719i 0.885385 + 0.464859i \(0.153895\pi\)
−0.885385 + 0.464859i \(0.846105\pi\)
\(458\) − 597.697i − 1.30501i
\(459\) 0 0
\(460\) −17.8873 −0.0388854
\(461\) 134.512i 0.291782i 0.989301 + 0.145891i \(0.0466049\pi\)
−0.989301 + 0.145891i \(0.953395\pi\)
\(462\) 0 0
\(463\) −186.040 −0.401815 −0.200908 0.979610i \(-0.564389\pi\)
−0.200908 + 0.979610i \(0.564389\pi\)
\(464\) 659.026i 1.42031i
\(465\) 0 0
\(466\) 92.5968 0.198706
\(467\) 583.197 1.24882 0.624408 0.781099i \(-0.285340\pi\)
0.624408 + 0.781099i \(0.285340\pi\)
\(468\) 0 0
\(469\) − 582.206i − 1.24138i
\(470\) − 574.561i − 1.22247i
\(471\) 0 0
\(472\) 458.440i 0.971271i
\(473\) 0 0
\(474\) 0 0
\(475\) − 649.359i − 1.36707i
\(476\) −25.4385 −0.0534422
\(477\) 0 0
\(478\) −28.4007 −0.0594158
\(479\) − 734.447i − 1.53329i −0.642070 0.766646i \(-0.721924\pi\)
0.642070 0.766646i \(-0.278076\pi\)
\(480\) 0 0
\(481\) 203.024i 0.422087i
\(482\) −174.033 −0.361065
\(483\) 0 0
\(484\) 0 0
\(485\) −240.705 −0.496300
\(486\) 0 0
\(487\) 289.221 0.593882 0.296941 0.954896i \(-0.404033\pi\)
0.296941 + 0.954896i \(0.404033\pi\)
\(488\) −794.814 −1.62872
\(489\) 0 0
\(490\) − 203.692i − 0.415699i
\(491\) 535.540i 1.09071i 0.838204 + 0.545356i \(0.183606\pi\)
−0.838204 + 0.545356i \(0.816394\pi\)
\(492\) 0 0
\(493\) 225.005 0.456400
\(494\) 135.076i 0.273432i
\(495\) 0 0
\(496\) 60.0012 0.120970
\(497\) − 271.449i − 0.546176i
\(498\) 0 0
\(499\) 446.596 0.894982 0.447491 0.894289i \(-0.352318\pi\)
0.447491 + 0.894289i \(0.352318\pi\)
\(500\) 43.1995 0.0863989
\(501\) 0 0
\(502\) 674.131i 1.34289i
\(503\) − 701.708i − 1.39505i −0.716563 0.697523i \(-0.754286\pi\)
0.716563 0.697523i \(-0.245714\pi\)
\(504\) 0 0
\(505\) − 642.641i − 1.27256i
\(506\) 0 0
\(507\) 0 0
\(508\) 155.822i 0.306736i
\(509\) −169.290 −0.332594 −0.166297 0.986076i \(-0.553181\pi\)
−0.166297 + 0.986076i \(0.553181\pi\)
\(510\) 0 0
\(511\) 383.174 0.749851
\(512\) − 329.793i − 0.644127i
\(513\) 0 0
\(514\) 53.5919i 0.104264i
\(515\) −585.400 −1.13670
\(516\) 0 0
\(517\) 0 0
\(518\) 827.596 1.59768
\(519\) 0 0
\(520\) 176.449 0.339325
\(521\) −666.820 −1.27989 −0.639943 0.768423i \(-0.721042\pi\)
−0.639943 + 0.768423i \(0.721042\pi\)
\(522\) 0 0
\(523\) − 253.003i − 0.483754i −0.970307 0.241877i \(-0.922237\pi\)
0.970307 0.241877i \(-0.0777630\pi\)
\(524\) 157.962i 0.301454i
\(525\) 0 0
\(526\) −409.507 −0.778530
\(527\) − 20.4857i − 0.0388722i
\(528\) 0 0
\(529\) −516.952 −0.977226
\(530\) 273.368i 0.515789i
\(531\) 0 0
\(532\) 79.4455 0.149334
\(533\) 102.531 0.192365
\(534\) 0 0
\(535\) 981.366i 1.83433i
\(536\) 691.274i 1.28969i
\(537\) 0 0
\(538\) 884.854i 1.64471i
\(539\) 0 0
\(540\) 0 0
\(541\) 383.554i 0.708971i 0.935061 + 0.354486i \(0.115344\pi\)
−0.935061 + 0.354486i \(0.884656\pi\)
\(542\) −53.0184 −0.0978199
\(543\) 0 0
\(544\) 66.5337 0.122305
\(545\) 1149.32i 2.10885i
\(546\) 0 0
\(547\) 376.753i 0.688762i 0.938830 + 0.344381i \(0.111911\pi\)
−0.938830 + 0.344381i \(0.888089\pi\)
\(548\) −97.9138 −0.178675
\(549\) 0 0
\(550\) 0 0
\(551\) −702.702 −1.27532
\(552\) 0 0
\(553\) −85.4116 −0.154451
\(554\) 174.091 0.314244
\(555\) 0 0
\(556\) − 13.8941i − 0.0249894i
\(557\) 810.940i 1.45591i 0.685627 + 0.727953i \(0.259528\pi\)
−0.685627 + 0.727953i \(0.740472\pi\)
\(558\) 0 0
\(559\) −141.129 −0.252466
\(560\) − 844.095i − 1.50731i
\(561\) 0 0
\(562\) 125.967 0.224140
\(563\) 62.2529i 0.110574i 0.998471 + 0.0552868i \(0.0176073\pi\)
−0.998471 + 0.0552868i \(0.982393\pi\)
\(564\) 0 0
\(565\) −785.713 −1.39064
\(566\) 269.228 0.475668
\(567\) 0 0
\(568\) 322.302i 0.567432i
\(569\) − 128.524i − 0.225878i −0.993602 0.112939i \(-0.963974\pi\)
0.993602 0.112939i \(-0.0360264\pi\)
\(570\) 0 0
\(571\) − 2.62521i − 0.00459757i −0.999997 0.00229879i \(-0.999268\pi\)
0.999997 0.00229879i \(-0.000731727\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) − 417.951i − 0.728138i
\(575\) −115.870 −0.201513
\(576\) 0 0
\(577\) −641.965 −1.11259 −0.556296 0.830984i \(-0.687778\pi\)
−0.556296 + 0.830984i \(0.687778\pi\)
\(578\) 540.957i 0.935911i
\(579\) 0 0
\(580\) − 186.167i − 0.320977i
\(581\) 536.596 0.923573
\(582\) 0 0
\(583\) 0 0
\(584\) −454.956 −0.779034
\(585\) 0 0
\(586\) 214.171 0.365479
\(587\) 570.395 0.971712 0.485856 0.874039i \(-0.338508\pi\)
0.485856 + 0.874039i \(0.338508\pi\)
\(588\) 0 0
\(589\) 63.9777i 0.108621i
\(590\) − 1053.32i − 1.78529i
\(591\) 0 0
\(592\) −1153.17 −1.94792
\(593\) 57.8790i 0.0976036i 0.998808 + 0.0488018i \(0.0155403\pi\)
−0.998808 + 0.0488018i \(0.984460\pi\)
\(594\) 0 0
\(595\) −288.192 −0.484356
\(596\) − 54.3775i − 0.0912375i
\(597\) 0 0
\(598\) 24.1026 0.0403054
\(599\) −288.442 −0.481539 −0.240769 0.970582i \(-0.577400\pi\)
−0.240769 + 0.970582i \(0.577400\pi\)
\(600\) 0 0
\(601\) 666.067i 1.10827i 0.832428 + 0.554133i \(0.186950\pi\)
−0.832428 + 0.554133i \(0.813050\pi\)
\(602\) 575.290i 0.955631i
\(603\) 0 0
\(604\) − 101.463i − 0.167985i
\(605\) 0 0
\(606\) 0 0
\(607\) 233.869i 0.385286i 0.981269 + 0.192643i \(0.0617059\pi\)
−0.981269 + 0.192643i \(0.938294\pi\)
\(608\) −207.788 −0.341756
\(609\) 0 0
\(610\) 1826.18 2.99374
\(611\) 111.706i 0.182825i
\(612\) 0 0
\(613\) 447.324i 0.729729i 0.931061 + 0.364865i \(0.118885\pi\)
−0.931061 + 0.364865i \(0.881115\pi\)
\(614\) −724.941 −1.18069
\(615\) 0 0
\(616\) 0 0
\(617\) 662.279 1.07339 0.536693 0.843778i \(-0.319673\pi\)
0.536693 + 0.843778i \(0.319673\pi\)
\(618\) 0 0
\(619\) −1177.52 −1.90229 −0.951145 0.308746i \(-0.900091\pi\)
−0.951145 + 0.308746i \(0.900091\pi\)
\(620\) −16.9496 −0.0273380
\(621\) 0 0
\(622\) − 555.753i − 0.893494i
\(623\) 311.242i 0.499586i
\(624\) 0 0
\(625\) −345.162 −0.552260
\(626\) 405.752i 0.648165i
\(627\) 0 0
\(628\) 54.9625 0.0875199
\(629\) 393.716i 0.625939i
\(630\) 0 0
\(631\) 663.644 1.05173 0.525867 0.850567i \(-0.323741\pi\)
0.525867 + 0.850567i \(0.323741\pi\)
\(632\) 101.412 0.160462
\(633\) 0 0
\(634\) − 702.434i − 1.10794i
\(635\) 1765.30i 2.78001i
\(636\) 0 0
\(637\) 39.6019i 0.0621693i
\(638\) 0 0
\(639\) 0 0
\(640\) 1150.44i 1.79756i
\(641\) −186.560 −0.291045 −0.145522 0.989355i \(-0.546486\pi\)
−0.145522 + 0.989355i \(0.546486\pi\)
\(642\) 0 0
\(643\) −41.4284 −0.0644298 −0.0322149 0.999481i \(-0.510256\pi\)
−0.0322149 + 0.999481i \(0.510256\pi\)
\(644\) − 14.1761i − 0.0220126i
\(645\) 0 0
\(646\) 261.947i 0.405491i
\(647\) 115.776 0.178943 0.0894716 0.995989i \(-0.471482\pi\)
0.0894716 + 0.995989i \(0.471482\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −231.812 −0.356634
\(651\) 0 0
\(652\) 15.5646 0.0238721
\(653\) −608.629 −0.932050 −0.466025 0.884772i \(-0.654314\pi\)
−0.466025 + 0.884772i \(0.654314\pi\)
\(654\) 0 0
\(655\) 1789.55i 2.73213i
\(656\) 582.371i 0.887760i
\(657\) 0 0
\(658\) 455.353 0.692026
\(659\) 725.000i 1.10015i 0.835114 + 0.550076i \(0.185401\pi\)
−0.835114 + 0.550076i \(0.814599\pi\)
\(660\) 0 0
\(661\) −781.088 −1.18168 −0.590838 0.806790i \(-0.701203\pi\)
−0.590838 + 0.806790i \(0.701203\pi\)
\(662\) 1295.59i 1.95708i
\(663\) 0 0
\(664\) −637.120 −0.959518
\(665\) 900.036 1.35344
\(666\) 0 0
\(667\) 125.389i 0.187989i
\(668\) 197.235i 0.295262i
\(669\) 0 0
\(670\) − 1588.29i − 2.37058i
\(671\) 0 0
\(672\) 0 0
\(673\) − 967.686i − 1.43787i −0.695078 0.718934i \(-0.744630\pi\)
0.695078 0.718934i \(-0.255370\pi\)
\(674\) 1038.55 1.54087
\(675\) 0 0
\(676\) −107.025 −0.158321
\(677\) 970.132i 1.43299i 0.697594 + 0.716494i \(0.254254\pi\)
−0.697594 + 0.716494i \(0.745746\pi\)
\(678\) 0 0
\(679\) − 190.765i − 0.280950i
\(680\) 342.181 0.503207
\(681\) 0 0
\(682\) 0 0
\(683\) −1104.74 −1.61749 −0.808745 0.588160i \(-0.799852\pi\)
−0.808745 + 0.588160i \(0.799852\pi\)
\(684\) 0 0
\(685\) −1109.26 −1.61936
\(686\) 802.960 1.17050
\(687\) 0 0
\(688\) − 801.606i − 1.16513i
\(689\) − 53.1482i − 0.0771382i
\(690\) 0 0
\(691\) 853.354 1.23496 0.617478 0.786588i \(-0.288155\pi\)
0.617478 + 0.786588i \(0.288155\pi\)
\(692\) − 143.180i − 0.206907i
\(693\) 0 0
\(694\) −45.7237 −0.0658842
\(695\) − 157.406i − 0.226484i
\(696\) 0 0
\(697\) 198.834 0.285271
\(698\) 781.901 1.12020
\(699\) 0 0
\(700\) 136.341i 0.194774i
\(701\) − 874.615i − 1.24767i −0.781557 0.623834i \(-0.785574\pi\)
0.781557 0.623834i \(-0.214426\pi\)
\(702\) 0 0
\(703\) − 1229.59i − 1.74906i
\(704\) 0 0
\(705\) 0 0
\(706\) − 303.656i − 0.430108i
\(707\) 509.309 0.720380
\(708\) 0 0
\(709\) 150.786 0.212674 0.106337 0.994330i \(-0.466088\pi\)
0.106337 + 0.994330i \(0.466088\pi\)
\(710\) − 740.528i − 1.04300i
\(711\) 0 0
\(712\) − 369.549i − 0.519030i
\(713\) 11.4160 0.0160113
\(714\) 0 0
\(715\) 0 0
\(716\) 35.6489 0.0497890
\(717\) 0 0
\(718\) 936.338 1.30409
\(719\) 731.799 1.01780 0.508900 0.860825i \(-0.330052\pi\)
0.508900 + 0.860825i \(0.330052\pi\)
\(720\) 0 0
\(721\) − 463.944i − 0.643473i
\(722\) − 37.5726i − 0.0520396i
\(723\) 0 0
\(724\) 17.7267 0.0244844
\(725\) − 1205.95i − 1.66338i
\(726\) 0 0
\(727\) −827.657 −1.13845 −0.569227 0.822180i \(-0.692758\pi\)
−0.569227 + 0.822180i \(0.692758\pi\)
\(728\) 139.840i 0.192088i
\(729\) 0 0
\(730\) 1045.32 1.43194
\(731\) −273.685 −0.374398
\(732\) 0 0
\(733\) − 76.3293i − 0.104133i −0.998644 0.0520664i \(-0.983419\pi\)
0.998644 0.0520664i \(-0.0165807\pi\)
\(734\) 47.7450i 0.0650477i
\(735\) 0 0
\(736\) 37.0772i 0.0503767i
\(737\) 0 0
\(738\) 0 0
\(739\) 1358.96i 1.83892i 0.393188 + 0.919458i \(0.371372\pi\)
−0.393188 + 0.919458i \(0.628628\pi\)
\(740\) 325.756 0.440210
\(741\) 0 0
\(742\) −216.651 −0.291982
\(743\) − 389.840i − 0.524684i −0.964975 0.262342i \(-0.915505\pi\)
0.964975 0.262342i \(-0.0844948\pi\)
\(744\) 0 0
\(745\) − 616.042i − 0.826902i
\(746\) 1049.51 1.40685
\(747\) 0 0
\(748\) 0 0
\(749\) −777.756 −1.03839
\(750\) 0 0
\(751\) 1254.95 1.67103 0.835517 0.549464i \(-0.185168\pi\)
0.835517 + 0.549464i \(0.185168\pi\)
\(752\) −634.487 −0.843733
\(753\) 0 0
\(754\) 250.854i 0.332698i
\(755\) − 1149.47i − 1.52248i
\(756\) 0 0
\(757\) 298.722 0.394613 0.197307 0.980342i \(-0.436781\pi\)
0.197307 + 0.980342i \(0.436781\pi\)
\(758\) − 1536.38i − 2.02689i
\(759\) 0 0
\(760\) −1068.65 −1.40611
\(761\) 575.246i 0.755908i 0.925824 + 0.377954i \(0.123372\pi\)
−0.925824 + 0.377954i \(0.876628\pi\)
\(762\) 0 0
\(763\) −910.865 −1.19379
\(764\) −77.6608 −0.101650
\(765\) 0 0
\(766\) − 151.712i − 0.198057i
\(767\) 204.787i 0.266997i
\(768\) 0 0
\(769\) 746.975i 0.971359i 0.874137 + 0.485680i \(0.161428\pi\)
−0.874137 + 0.485680i \(0.838572\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 26.0788i 0.0337808i
\(773\) −4.05557 −0.00524654 −0.00262327 0.999997i \(-0.500835\pi\)
−0.00262327 + 0.999997i \(0.500835\pi\)
\(774\) 0 0
\(775\) −109.796 −0.141672
\(776\) 226.502i 0.291884i
\(777\) 0 0
\(778\) 1088.17i 1.39868i
\(779\) −620.967 −0.797133
\(780\) 0 0
\(781\) 0 0
\(782\) 46.7413 0.0597714
\(783\) 0 0
\(784\) −224.937 −0.286910
\(785\) 622.668 0.793208
\(786\) 0 0
\(787\) 59.5811i 0.0757066i 0.999283 + 0.0378533i \(0.0120520\pi\)
−0.999283 + 0.0378533i \(0.987948\pi\)
\(788\) − 2.30944i − 0.00293076i
\(789\) 0 0
\(790\) −233.007 −0.294946
\(791\) − 622.697i − 0.787227i
\(792\) 0 0
\(793\) −355.047 −0.447726
\(794\) − 312.759i − 0.393903i
\(795\) 0 0
\(796\) 123.478 0.155124
\(797\) 187.364 0.235086 0.117543 0.993068i \(-0.462498\pi\)
0.117543 + 0.993068i \(0.462498\pi\)
\(798\) 0 0
\(799\) 216.627i 0.271123i
\(800\) − 356.598i − 0.445747i
\(801\) 0 0
\(802\) − 1293.15i − 1.61241i
\(803\) 0 0
\(804\) 0 0
\(805\) − 160.601i − 0.199504i
\(806\) 22.8391 0.0283364
\(807\) 0 0
\(808\) −604.721 −0.748417
\(809\) 1451.43i 1.79411i 0.441923 + 0.897053i \(0.354297\pi\)
−0.441923 + 0.897053i \(0.645703\pi\)
\(810\) 0 0
\(811\) 1043.47i 1.28665i 0.765594 + 0.643324i \(0.222445\pi\)
−0.765594 + 0.643324i \(0.777555\pi\)
\(812\) 147.541 0.181701
\(813\) 0 0
\(814\) 0 0
\(815\) 176.331 0.216357
\(816\) 0 0
\(817\) 854.731 1.04618
\(818\) −452.058 −0.552638
\(819\) 0 0
\(820\) − 164.512i − 0.200625i
\(821\) 165.085i 0.201078i 0.994933 + 0.100539i \(0.0320568\pi\)
−0.994933 + 0.100539i \(0.967943\pi\)
\(822\) 0 0
\(823\) −118.929 −0.144507 −0.0722535 0.997386i \(-0.523019\pi\)
−0.0722535 + 0.997386i \(0.523019\pi\)
\(824\) 550.857i 0.668516i
\(825\) 0 0
\(826\) 834.784 1.01063
\(827\) 1445.00i 1.74728i 0.486569 + 0.873642i \(0.338248\pi\)
−0.486569 + 0.873642i \(0.661752\pi\)
\(828\) 0 0
\(829\) −504.206 −0.608210 −0.304105 0.952639i \(-0.598357\pi\)
−0.304105 + 0.952639i \(0.598357\pi\)
\(830\) 1463.86 1.76369
\(831\) 0 0
\(832\) − 160.193i − 0.192540i
\(833\) 76.7983i 0.0921949i
\(834\) 0 0
\(835\) 2234.47i 2.67601i
\(836\) 0 0
\(837\) 0 0
\(838\) 368.216i 0.439399i
\(839\) −396.145 −0.472164 −0.236082 0.971733i \(-0.575863\pi\)
−0.236082 + 0.971733i \(0.575863\pi\)
\(840\) 0 0
\(841\) −464.016 −0.551743
\(842\) 652.647i 0.775115i
\(843\) 0 0
\(844\) 183.317i 0.217200i
\(845\) −1212.48 −1.43489
\(846\) 0 0
\(847\) 0 0
\(848\) 301.880 0.355991
\(849\) 0 0
\(850\) −449.544 −0.528875
\(851\) −219.406 −0.257821
\(852\) 0 0
\(853\) 56.0848i 0.0657500i 0.999459 + 0.0328750i \(0.0104663\pi\)
−0.999459 + 0.0328750i \(0.989534\pi\)
\(854\) 1447.29i 1.69472i
\(855\) 0 0
\(856\) 923.458 1.07881
\(857\) 108.138i 0.126182i 0.998008 + 0.0630911i \(0.0200959\pi\)
−0.998008 + 0.0630911i \(0.979904\pi\)
\(858\) 0 0
\(859\) 910.177 1.05958 0.529789 0.848130i \(-0.322271\pi\)
0.529789 + 0.848130i \(0.322271\pi\)
\(860\) 226.444i 0.263307i
\(861\) 0 0
\(862\) 901.314 1.04561
\(863\) −979.204 −1.13465 −0.567326 0.823494i \(-0.692022\pi\)
−0.567326 + 0.823494i \(0.692022\pi\)
\(864\) 0 0
\(865\) − 1622.08i − 1.87524i
\(866\) − 642.074i − 0.741425i
\(867\) 0 0
\(868\) − 13.4330i − 0.0154758i
\(869\) 0 0
\(870\) 0 0
\(871\) 308.795i 0.354529i
\(872\) 1081.50 1.24026
\(873\) 0 0
\(874\) −145.975 −0.167020
\(875\) 387.866i 0.443275i
\(876\) 0 0
\(877\) − 544.429i − 0.620785i −0.950608 0.310393i \(-0.899539\pi\)
0.950608 0.310393i \(-0.100461\pi\)
\(878\) −1493.87 −1.70145
\(879\) 0 0
\(880\) 0 0
\(881\) 1232.58 1.39907 0.699534 0.714600i \(-0.253391\pi\)
0.699534 + 0.714600i \(0.253391\pi\)
\(882\) 0 0
\(883\) 762.570 0.863613 0.431807 0.901966i \(-0.357876\pi\)
0.431807 + 0.901966i \(0.357876\pi\)
\(884\) 13.4923 0.0152627
\(885\) 0 0
\(886\) 1662.69i 1.87662i
\(887\) − 361.349i − 0.407384i −0.979035 0.203692i \(-0.934706\pi\)
0.979035 0.203692i \(-0.0652940\pi\)
\(888\) 0 0
\(889\) −1399.05 −1.57373
\(890\) 849.085i 0.954028i
\(891\) 0 0
\(892\) −30.1800 −0.0338340
\(893\) − 676.537i − 0.757600i
\(894\) 0 0
\(895\) 403.866 0.451247
\(896\) −911.749 −1.01758
\(897\) 0 0
\(898\) 757.025i 0.843012i
\(899\) 118.816i 0.132164i
\(900\) 0 0
\(901\) − 103.068i − 0.114393i
\(902\) 0 0
\(903\) 0 0
\(904\) 739.351i 0.817866i
\(905\) 200.825 0.221906
\(906\) 0 0
\(907\) 1330.09 1.46647 0.733234 0.679976i \(-0.238010\pi\)
0.733234 + 0.679976i \(0.238010\pi\)
\(908\) − 88.4304i − 0.0973903i
\(909\) 0 0
\(910\) − 321.300i − 0.353077i
\(911\) 0.221374 0.000243001 0 0.000121501 1.00000i \(-0.499961\pi\)
0.000121501 1.00000i \(0.499961\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −918.614 −1.00505
\(915\) 0 0
\(916\) 186.452 0.203550
\(917\) −1418.26 −1.54663
\(918\) 0 0
\(919\) 443.268i 0.482338i 0.970483 + 0.241169i \(0.0775307\pi\)
−0.970483 + 0.241169i \(0.922469\pi\)
\(920\) 190.687i 0.207268i
\(921\) 0 0
\(922\) −290.821 −0.315424
\(923\) 143.974i 0.155984i
\(924\) 0 0
\(925\) 2110.18 2.28128
\(926\) − 402.229i − 0.434372i
\(927\) 0 0
\(928\) −385.891 −0.415831
\(929\) −288.120 −0.310140 −0.155070 0.987904i \(-0.549560\pi\)
−0.155070 + 0.987904i \(0.549560\pi\)
\(930\) 0 0
\(931\) − 239.845i − 0.257621i
\(932\) 28.8856i 0.0309932i
\(933\) 0 0
\(934\) 1260.90i 1.35000i
\(935\) 0 0
\(936\) 0 0
\(937\) − 1462.84i − 1.56120i −0.625033 0.780599i \(-0.714914\pi\)
0.625033 0.780599i \(-0.285086\pi\)
\(938\) 1258.76 1.34196
\(939\) 0 0
\(940\) 179.235 0.190675
\(941\) − 1069.21i − 1.13624i −0.822944 0.568122i \(-0.807670\pi\)
0.822944 0.568122i \(-0.192330\pi\)
\(942\) 0 0
\(943\) 110.804i 0.117502i
\(944\) −1163.18 −1.23219
\(945\) 0 0
\(946\) 0 0
\(947\) −356.893 −0.376867 −0.188433 0.982086i \(-0.560341\pi\)
−0.188433 + 0.982086i \(0.560341\pi\)
\(948\) 0 0
\(949\) −203.231 −0.214153
\(950\) 1403.95 1.47784
\(951\) 0 0
\(952\) 271.186i 0.284860i
\(953\) − 1072.25i − 1.12513i −0.826753 0.562565i \(-0.809815\pi\)
0.826753 0.562565i \(-0.190185\pi\)
\(954\) 0 0
\(955\) −879.817 −0.921275
\(956\) − 8.85963i − 0.00926740i
\(957\) 0 0
\(958\) 1587.91 1.65753
\(959\) − 879.118i − 0.916703i
\(960\) 0 0
\(961\) −950.182 −0.988743
\(962\) −438.947 −0.456286
\(963\) 0 0
\(964\) − 54.2898i − 0.0563172i
\(965\) 295.446i 0.306161i
\(966\) 0 0
\(967\) 273.281i 0.282607i 0.989966 + 0.141303i \(0.0451293\pi\)
−0.989966 + 0.141303i \(0.954871\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) − 520.417i − 0.536512i
\(971\) 644.921 0.664182 0.332091 0.943247i \(-0.392246\pi\)
0.332091 + 0.943247i \(0.392246\pi\)
\(972\) 0 0
\(973\) 124.748 0.128210
\(974\) 625.309i 0.642001i
\(975\) 0 0
\(976\) − 2016.65i − 2.06624i
\(977\) 1637.62 1.67617 0.838086 0.545539i \(-0.183675\pi\)
0.838086 + 0.545539i \(0.183675\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 63.5420 0.0648388
\(981\) 0 0
\(982\) −1157.86 −1.17909
\(983\) 913.097 0.928888 0.464444 0.885602i \(-0.346254\pi\)
0.464444 + 0.885602i \(0.346254\pi\)
\(984\) 0 0
\(985\) − 26.1636i − 0.0265620i
\(986\) 486.472i 0.493379i
\(987\) 0 0
\(988\) −42.1370 −0.0426487
\(989\) − 152.516i − 0.154213i
\(990\) 0 0
\(991\) −1447.57 −1.46072 −0.730360 0.683062i \(-0.760648\pi\)
−0.730360 + 0.683062i \(0.760648\pi\)
\(992\) 35.1336i 0.0354169i
\(993\) 0 0
\(994\) 586.887 0.590429
\(995\) 1398.88 1.40591
\(996\) 0 0
\(997\) 909.292i 0.912028i 0.889973 + 0.456014i \(0.150723\pi\)
−0.889973 + 0.456014i \(0.849277\pi\)
\(998\) 965.562i 0.967497i
\(999\) 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 1089.3.c.k.604.6 8
3.2 odd 2 121.3.b.c.120.3 8
11.10 odd 2 inner 1089.3.c.k.604.3 8
33.2 even 10 121.3.d.f.40.6 32
33.5 odd 10 121.3.d.f.118.6 32
33.8 even 10 121.3.d.f.112.6 32
33.14 odd 10 121.3.d.f.112.3 32
33.17 even 10 121.3.d.f.118.3 32
33.20 odd 10 121.3.d.f.40.3 32
33.26 odd 10 121.3.d.f.94.6 32
33.29 even 10 121.3.d.f.94.3 32
33.32 even 2 121.3.b.c.120.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.3.b.c.120.3 8 3.2 odd 2
121.3.b.c.120.6 yes 8 33.32 even 2
121.3.d.f.40.3 32 33.20 odd 10
121.3.d.f.40.6 32 33.2 even 10
121.3.d.f.94.3 32 33.29 even 10
121.3.d.f.94.6 32 33.26 odd 10
121.3.d.f.112.3 32 33.14 odd 10
121.3.d.f.112.6 32 33.8 even 10
121.3.d.f.118.3 32 33.17 even 10
121.3.d.f.118.6 32 33.5 odd 10
1089.3.c.k.604.3 8 11.10 odd 2 inner
1089.3.c.k.604.6 8 1.1 even 1 trivial