Properties

Label 1890.2.g.a.379.1
Level $1890$
Weight $2$
Character 1890.379
Analytic conductor $15.092$
Analytic rank $0$
Dimension $2$
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] = [1890,2,Mod(379,1890)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1890, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1890.379");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1890 = 2 \cdot 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1890.g (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: \(15.0917259820\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 379.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1890.379
Dual form 1890.2.g.a.379.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +(-2.00000 - 1.00000i) q^{5} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} +(-2.00000 - 1.00000i) q^{5} +1.00000i q^{7} +1.00000i q^{8} +(-1.00000 + 2.00000i) q^{10} -4.00000 q^{11} -6.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -4.00000i q^{17} +(2.00000 + 1.00000i) q^{20} +4.00000i q^{22} +4.00000i q^{23} +(3.00000 + 4.00000i) q^{25} -6.00000 q^{26} -1.00000i q^{28} -3.00000 q^{29} +7.00000 q^{31} -1.00000i q^{32} -4.00000 q^{34} +(1.00000 - 2.00000i) q^{35} -1.00000i q^{37} +(1.00000 - 2.00000i) q^{40} -7.00000 q^{41} +10.0000i q^{43} +4.00000 q^{44} +4.00000 q^{46} +13.0000i q^{47} -1.00000 q^{49} +(4.00000 - 3.00000i) q^{50} +6.00000i q^{52} +6.00000i q^{53} +(8.00000 + 4.00000i) q^{55} -1.00000 q^{56} +3.00000i q^{58} +5.00000 q^{59} -7.00000 q^{61} -7.00000i q^{62} -1.00000 q^{64} +(-6.00000 + 12.0000i) q^{65} +4.00000i q^{67} +4.00000i q^{68} +(-2.00000 - 1.00000i) q^{70} -3.00000 q^{71} +7.00000i q^{73} -1.00000 q^{74} -4.00000i q^{77} -12.0000 q^{79} +(-2.00000 - 1.00000i) q^{80} +7.00000i q^{82} -2.00000i q^{83} +(-4.00000 + 8.00000i) q^{85} +10.0000 q^{86} -4.00000i q^{88} -14.0000 q^{89} +6.00000 q^{91} -4.00000i q^{92} +13.0000 q^{94} +10.0000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 4 q^{5} - 2 q^{10} - 8 q^{11} + 2 q^{14} + 2 q^{16} + 4 q^{20} + 6 q^{25} - 12 q^{26} - 6 q^{29} + 14 q^{31} - 8 q^{34} + 2 q^{35} + 2 q^{40} - 14 q^{41} + 8 q^{44} + 8 q^{46} - 2 q^{49} + 8 q^{50} + 16 q^{55} - 2 q^{56} + 10 q^{59} - 14 q^{61} - 2 q^{64} - 12 q^{65} - 4 q^{70} - 6 q^{71} - 2 q^{74} - 24 q^{79} - 4 q^{80} - 8 q^{85} + 20 q^{86} - 28 q^{89} + 12 q^{91} + 26 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1890\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(1081\) \(1541\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −2.00000 1.00000i −0.894427 0.447214i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) −1.00000 + 2.00000i −0.316228 + 0.632456i
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 6.00000i 1.66410i −0.554700 0.832050i \(-0.687167\pi\)
0.554700 0.832050i \(-0.312833\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000i 0.970143i −0.874475 0.485071i \(-0.838794\pi\)
0.874475 0.485071i \(-0.161206\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 2.00000 + 1.00000i 0.447214 + 0.223607i
\(21\) 0 0
\(22\) 4.00000i 0.852803i
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) 3.00000 + 4.00000i 0.600000 + 0.800000i
\(26\) −6.00000 −1.17670
\(27\) 0 0
\(28\) 1.00000i 0.188982i
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) 1.00000 2.00000i 0.169031 0.338062i
\(36\) 0 0
\(37\) 1.00000i 0.164399i −0.996616 0.0821995i \(-0.973806\pi\)
0.996616 0.0821995i \(-0.0261945\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 1.00000 2.00000i 0.158114 0.316228i
\(41\) −7.00000 −1.09322 −0.546608 0.837389i \(-0.684081\pi\)
−0.546608 + 0.837389i \(0.684081\pi\)
\(42\) 0 0
\(43\) 10.0000i 1.52499i 0.646997 + 0.762493i \(0.276025\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 13.0000i 1.89624i 0.317905 + 0.948122i \(0.397021\pi\)
−0.317905 + 0.948122i \(0.602979\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 4.00000 3.00000i 0.565685 0.424264i
\(51\) 0 0
\(52\) 6.00000i 0.832050i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 8.00000 + 4.00000i 1.07872 + 0.539360i
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 3.00000i 0.393919i
\(59\) 5.00000 0.650945 0.325472 0.945552i \(-0.394477\pi\)
0.325472 + 0.945552i \(0.394477\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 7.00000i 0.889001i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) −6.00000 + 12.0000i −0.744208 + 1.48842i
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 4.00000i 0.485071i
\(69\) 0 0
\(70\) −2.00000 1.00000i −0.239046 0.119523i
\(71\) −3.00000 −0.356034 −0.178017 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) 7.00000i 0.819288i 0.912245 + 0.409644i \(0.134347\pi\)
−0.912245 + 0.409644i \(0.865653\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) 0 0
\(77\) 4.00000i 0.455842i
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) −2.00000 1.00000i −0.223607 0.111803i
\(81\) 0 0
\(82\) 7.00000i 0.773021i
\(83\) 2.00000i 0.219529i −0.993958 0.109764i \(-0.964990\pi\)
0.993958 0.109764i \(-0.0350096\pi\)
\(84\) 0 0
\(85\) −4.00000 + 8.00000i −0.433861 + 0.867722i
\(86\) 10.0000 1.07833
\(87\) 0 0
\(88\) 4.00000i 0.426401i
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 4.00000i 0.417029i
\(93\) 0 0
\(94\) 13.0000 1.34085
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 1.00000i 0.101015i
\(99\) 0 0
\(100\) −3.00000 4.00000i −0.300000 0.400000i
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) 14.0000i 1.37946i 0.724066 + 0.689730i \(0.242271\pi\)
−0.724066 + 0.689730i \(0.757729\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 9.00000i 0.870063i −0.900415 0.435031i \(-0.856737\pi\)
0.900415 0.435031i \(-0.143263\pi\)
\(108\) 0 0
\(109\) 20.0000 1.91565 0.957826 0.287348i \(-0.0927736\pi\)
0.957826 + 0.287348i \(0.0927736\pi\)
\(110\) 4.00000 8.00000i 0.381385 0.762770i
\(111\) 0 0
\(112\) 1.00000i 0.0944911i
\(113\) 3.00000i 0.282216i 0.989994 + 0.141108i \(0.0450665\pi\)
−0.989994 + 0.141108i \(0.954933\pi\)
\(114\) 0 0
\(115\) 4.00000 8.00000i 0.373002 0.746004i
\(116\) 3.00000 0.278543
\(117\) 0 0
\(118\) 5.00000i 0.460287i
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 7.00000i 0.633750i
\(123\) 0 0
\(124\) −7.00000 −0.628619
\(125\) −2.00000 11.0000i −0.178885 0.983870i
\(126\) 0 0
\(127\) 7.00000i 0.621150i −0.950549 0.310575i \(-0.899478\pi\)
0.950549 0.310575i \(-0.100522\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 12.0000 + 6.00000i 1.05247 + 0.526235i
\(131\) 15.0000 1.31056 0.655278 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 4.00000 0.342997
\(137\) 15.0000i 1.28154i −0.767734 0.640768i \(-0.778616\pi\)
0.767734 0.640768i \(-0.221384\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) −1.00000 + 2.00000i −0.0845154 + 0.169031i
\(141\) 0 0
\(142\) 3.00000i 0.251754i
\(143\) 24.0000i 2.00698i
\(144\) 0 0
\(145\) 6.00000 + 3.00000i 0.498273 + 0.249136i
\(146\) 7.00000 0.579324
\(147\) 0 0
\(148\) 1.00000i 0.0821995i
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −14.0000 −1.13930 −0.569652 0.821886i \(-0.692922\pi\)
−0.569652 + 0.821886i \(0.692922\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −4.00000 −0.322329
\(155\) −14.0000 7.00000i −1.12451 0.562254i
\(156\) 0 0
\(157\) 14.0000i 1.11732i −0.829396 0.558661i \(-0.811315\pi\)
0.829396 0.558661i \(-0.188685\pi\)
\(158\) 12.0000i 0.954669i
\(159\) 0 0
\(160\) −1.00000 + 2.00000i −0.0790569 + 0.158114i
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) 14.0000i 1.09656i 0.836293 + 0.548282i \(0.184718\pi\)
−0.836293 + 0.548282i \(0.815282\pi\)
\(164\) 7.00000 0.546608
\(165\) 0 0
\(166\) −2.00000 −0.155230
\(167\) 3.00000i 0.232147i −0.993241 0.116073i \(-0.962969\pi\)
0.993241 0.116073i \(-0.0370308\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 8.00000 + 4.00000i 0.613572 + 0.306786i
\(171\) 0 0
\(172\) 10.0000i 0.762493i
\(173\) 17.0000i 1.29249i 0.763132 + 0.646243i \(0.223661\pi\)
−0.763132 + 0.646243i \(0.776339\pi\)
\(174\) 0 0
\(175\) −4.00000 + 3.00000i −0.302372 + 0.226779i
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) 14.0000i 1.04934i
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 0 0
\(181\) 23.0000 1.70958 0.854788 0.518977i \(-0.173687\pi\)
0.854788 + 0.518977i \(0.173687\pi\)
\(182\) 6.00000i 0.444750i
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) −1.00000 + 2.00000i −0.0735215 + 0.147043i
\(186\) 0 0
\(187\) 16.0000i 1.17004i
\(188\) 13.0000i 0.948122i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 2.00000i 0.143963i −0.997406 0.0719816i \(-0.977068\pi\)
0.997406 0.0719816i \(-0.0229323\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 8.00000i 0.569976i −0.958531 0.284988i \(-0.908010\pi\)
0.958531 0.284988i \(-0.0919897\pi\)
\(198\) 0 0
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) −4.00000 + 3.00000i −0.282843 + 0.212132i
\(201\) 0 0
\(202\) 10.0000i 0.703598i
\(203\) 3.00000i 0.210559i
\(204\) 0 0
\(205\) 14.0000 + 7.00000i 0.977802 + 0.488901i
\(206\) 14.0000 0.975426
\(207\) 0 0
\(208\) 6.00000i 0.416025i
\(209\) 0 0
\(210\) 0 0
\(211\) −15.0000 −1.03264 −0.516321 0.856395i \(-0.672699\pi\)
−0.516321 + 0.856395i \(0.672699\pi\)
\(212\) 6.00000i 0.412082i
\(213\) 0 0
\(214\) −9.00000 −0.615227
\(215\) 10.0000 20.0000i 0.681994 1.36399i
\(216\) 0 0
\(217\) 7.00000i 0.475191i
\(218\) 20.0000i 1.35457i
\(219\) 0 0
\(220\) −8.00000 4.00000i −0.539360 0.269680i
\(221\) −24.0000 −1.61441
\(222\) 0 0
\(223\) 4.00000i 0.267860i 0.990991 + 0.133930i \(0.0427597\pi\)
−0.990991 + 0.133930i \(0.957240\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 3.00000 0.199557
\(227\) 16.0000i 1.06196i −0.847385 0.530979i \(-0.821824\pi\)
0.847385 0.530979i \(-0.178176\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) −8.00000 4.00000i −0.527504 0.263752i
\(231\) 0 0
\(232\) 3.00000i 0.196960i
\(233\) 11.0000i 0.720634i 0.932830 + 0.360317i \(0.117331\pi\)
−0.932830 + 0.360317i \(0.882669\pi\)
\(234\) 0 0
\(235\) 13.0000 26.0000i 0.848026 1.69605i
\(236\) −5.00000 −0.325472
\(237\) 0 0
\(238\) 4.00000i 0.259281i
\(239\) 13.0000 0.840900 0.420450 0.907316i \(-0.361872\pi\)
0.420450 + 0.907316i \(0.361872\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 5.00000i 0.321412i
\(243\) 0 0
\(244\) 7.00000 0.448129
\(245\) 2.00000 + 1.00000i 0.127775 + 0.0638877i
\(246\) 0 0
\(247\) 0 0
\(248\) 7.00000i 0.444500i
\(249\) 0 0
\(250\) −11.0000 + 2.00000i −0.695701 + 0.126491i
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 16.0000i 1.00591i
\(254\) −7.00000 −0.439219
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 24.0000i 1.49708i 0.663090 + 0.748539i \(0.269245\pi\)
−0.663090 + 0.748539i \(0.730755\pi\)
\(258\) 0 0
\(259\) 1.00000 0.0621370
\(260\) 6.00000 12.0000i 0.372104 0.744208i
\(261\) 0 0
\(262\) 15.0000i 0.926703i
\(263\) 16.0000i 0.986602i 0.869859 + 0.493301i \(0.164210\pi\)
−0.869859 + 0.493301i \(0.835790\pi\)
\(264\) 0 0
\(265\) 6.00000 12.0000i 0.368577 0.737154i
\(266\) 0 0
\(267\) 0 0
\(268\) 4.00000i 0.244339i
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −11.0000 −0.668202 −0.334101 0.942537i \(-0.608433\pi\)
−0.334101 + 0.942537i \(0.608433\pi\)
\(272\) 4.00000i 0.242536i
\(273\) 0 0
\(274\) −15.0000 −0.906183
\(275\) −12.0000 16.0000i −0.723627 0.964836i
\(276\) 0 0
\(277\) 2.00000i 0.120168i 0.998193 + 0.0600842i \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) 14.0000i 0.839664i
\(279\) 0 0
\(280\) 2.00000 + 1.00000i 0.119523 + 0.0597614i
\(281\) 24.0000 1.43172 0.715860 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) 0 0
\(283\) 7.00000i 0.416107i −0.978117 0.208053i \(-0.933287\pi\)
0.978117 0.208053i \(-0.0667128\pi\)
\(284\) 3.00000 0.178017
\(285\) 0 0
\(286\) 24.0000 1.41915
\(287\) 7.00000i 0.413197i
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 3.00000 6.00000i 0.176166 0.352332i
\(291\) 0 0
\(292\) 7.00000i 0.409644i
\(293\) 15.0000i 0.876309i 0.898900 + 0.438155i \(0.144368\pi\)
−0.898900 + 0.438155i \(0.855632\pi\)
\(294\) 0 0
\(295\) −10.0000 5.00000i −0.582223 0.291111i
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) 10.0000i 0.579284i
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) −10.0000 −0.576390
\(302\) 14.0000i 0.805609i
\(303\) 0 0
\(304\) 0 0
\(305\) 14.0000 + 7.00000i 0.801638 + 0.400819i
\(306\) 0 0
\(307\) 12.0000i 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 4.00000i 0.227921i
\(309\) 0 0
\(310\) −7.00000 + 14.0000i −0.397573 + 0.795147i
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) 0 0
\(313\) 26.0000i 1.46961i −0.678280 0.734803i \(-0.737274\pi\)
0.678280 0.734803i \(-0.262726\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 12.0000 0.675053
\(317\) 14.0000i 0.786318i 0.919470 + 0.393159i \(0.128618\pi\)
−0.919470 + 0.393159i \(0.871382\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) 2.00000 + 1.00000i 0.111803 + 0.0559017i
\(321\) 0 0
\(322\) 4.00000i 0.222911i
\(323\) 0 0
\(324\) 0 0
\(325\) 24.0000 18.0000i 1.33128 0.998460i
\(326\) 14.0000 0.775388
\(327\) 0 0
\(328\) 7.00000i 0.386510i
\(329\) −13.0000 −0.716713
\(330\) 0 0
\(331\) −35.0000 −1.92377 −0.961887 0.273447i \(-0.911836\pi\)
−0.961887 + 0.273447i \(0.911836\pi\)
\(332\) 2.00000i 0.109764i
\(333\) 0 0
\(334\) −3.00000 −0.164153
\(335\) 4.00000 8.00000i 0.218543 0.437087i
\(336\) 0 0
\(337\) 16.0000i 0.871576i −0.900049 0.435788i \(-0.856470\pi\)
0.900049 0.435788i \(-0.143530\pi\)
\(338\) 23.0000i 1.25104i
\(339\) 0 0
\(340\) 4.00000 8.00000i 0.216930 0.433861i
\(341\) −28.0000 −1.51629
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) −10.0000 −0.539164
\(345\) 0 0
\(346\) 17.0000 0.913926
\(347\) 27.0000i 1.44944i −0.689046 0.724718i \(-0.741970\pi\)
0.689046 0.724718i \(-0.258030\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 3.00000 + 4.00000i 0.160357 + 0.213809i
\(351\) 0 0
\(352\) 4.00000i 0.213201i
\(353\) 32.0000i 1.70319i 0.524202 + 0.851594i \(0.324364\pi\)
−0.524202 + 0.851594i \(0.675636\pi\)
\(354\) 0 0
\(355\) 6.00000 + 3.00000i 0.318447 + 0.159223i
\(356\) 14.0000 0.741999
\(357\) 0 0
\(358\) 16.0000i 0.845626i
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 23.0000i 1.20885i
\(363\) 0 0
\(364\) −6.00000 −0.314485
\(365\) 7.00000 14.0000i 0.366397 0.732793i
\(366\) 0 0
\(367\) 2.00000i 0.104399i 0.998637 + 0.0521996i \(0.0166232\pi\)
−0.998637 + 0.0521996i \(0.983377\pi\)
\(368\) 4.00000i 0.208514i
\(369\) 0 0
\(370\) 2.00000 + 1.00000i 0.103975 + 0.0519875i
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) 3.00000i 0.155334i −0.996979 0.0776671i \(-0.975253\pi\)
0.996979 0.0776671i \(-0.0247471\pi\)
\(374\) 16.0000 0.827340
\(375\) 0 0
\(376\) −13.0000 −0.670424
\(377\) 18.0000i 0.927047i
\(378\) 0 0
\(379\) 1.00000 0.0513665 0.0256833 0.999670i \(-0.491824\pi\)
0.0256833 + 0.999670i \(0.491824\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 8.00000i 0.408781i 0.978889 + 0.204390i \(0.0655212\pi\)
−0.978889 + 0.204390i \(0.934479\pi\)
\(384\) 0 0
\(385\) −4.00000 + 8.00000i −0.203859 + 0.407718i
\(386\) −2.00000 −0.101797
\(387\) 0 0
\(388\) 10.0000i 0.507673i
\(389\) −23.0000 −1.16615 −0.583073 0.812420i \(-0.698150\pi\)
−0.583073 + 0.812420i \(0.698150\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 1.00000i 0.0505076i
\(393\) 0 0
\(394\) −8.00000 −0.403034
\(395\) 24.0000 + 12.0000i 1.20757 + 0.603786i
\(396\) 0 0
\(397\) 14.0000i 0.702640i 0.936255 + 0.351320i \(0.114267\pi\)
−0.936255 + 0.351320i \(0.885733\pi\)
\(398\) 24.0000i 1.20301i
\(399\) 0 0
\(400\) 3.00000 + 4.00000i 0.150000 + 0.200000i
\(401\) 16.0000 0.799002 0.399501 0.916733i \(-0.369183\pi\)
0.399501 + 0.916733i \(0.369183\pi\)
\(402\) 0 0
\(403\) 42.0000i 2.09217i
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) −3.00000 −0.148888
\(407\) 4.00000i 0.198273i
\(408\) 0 0
\(409\) −20.0000 −0.988936 −0.494468 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(410\) 7.00000 14.0000i 0.345705 0.691411i
\(411\) 0 0
\(412\) 14.0000i 0.689730i
\(413\) 5.00000i 0.246034i
\(414\) 0 0
\(415\) −2.00000 + 4.00000i −0.0981761 + 0.196352i
\(416\) −6.00000 −0.294174
\(417\) 0 0
\(418\) 0 0
\(419\) 25.0000 1.22133 0.610665 0.791889i \(-0.290902\pi\)
0.610665 + 0.791889i \(0.290902\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) 15.0000i 0.730189i
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 16.0000 12.0000i 0.776114 0.582086i
\(426\) 0 0
\(427\) 7.00000i 0.338754i
\(428\) 9.00000i 0.435031i
\(429\) 0 0
\(430\) −20.0000 10.0000i −0.964486 0.482243i
\(431\) 11.0000 0.529851 0.264926 0.964269i \(-0.414653\pi\)
0.264926 + 0.964269i \(0.414653\pi\)
\(432\) 0 0
\(433\) 29.0000i 1.39365i −0.717241 0.696826i \(-0.754595\pi\)
0.717241 0.696826i \(-0.245405\pi\)
\(434\) 7.00000 0.336011
\(435\) 0 0
\(436\) −20.0000 −0.957826
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) −4.00000 + 8.00000i −0.190693 + 0.381385i
\(441\) 0 0
\(442\) 24.0000i 1.14156i
\(443\) 9.00000i 0.427603i 0.976877 + 0.213801i \(0.0685846\pi\)
−0.976877 + 0.213801i \(0.931415\pi\)
\(444\) 0 0
\(445\) 28.0000 + 14.0000i 1.32733 + 0.663664i
\(446\) 4.00000 0.189405
\(447\) 0 0
\(448\) 1.00000i 0.0472456i
\(449\) −36.0000 −1.69895 −0.849473 0.527633i \(-0.823080\pi\)
−0.849473 + 0.527633i \(0.823080\pi\)
\(450\) 0 0
\(451\) 28.0000 1.31847
\(452\) 3.00000i 0.141108i
\(453\) 0 0
\(454\) −16.0000 −0.750917
\(455\) −12.0000 6.00000i −0.562569 0.281284i
\(456\) 0 0
\(457\) 8.00000i 0.374224i 0.982339 + 0.187112i \(0.0599128\pi\)
−0.982339 + 0.187112i \(0.940087\pi\)
\(458\) 14.0000i 0.654177i
\(459\) 0 0
\(460\) −4.00000 + 8.00000i −0.186501 + 0.373002i
\(461\) −36.0000 −1.67669 −0.838344 0.545142i \(-0.816476\pi\)
−0.838344 + 0.545142i \(0.816476\pi\)
\(462\) 0 0
\(463\) 33.0000i 1.53364i 0.641862 + 0.766820i \(0.278162\pi\)
−0.641862 + 0.766820i \(0.721838\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) 11.0000 0.509565
\(467\) 18.0000i 0.832941i 0.909149 + 0.416470i \(0.136733\pi\)
−0.909149 + 0.416470i \(0.863267\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) −26.0000 13.0000i −1.19929 0.599645i
\(471\) 0 0
\(472\) 5.00000i 0.230144i
\(473\) 40.0000i 1.83920i
\(474\) 0 0
\(475\) 0 0
\(476\) −4.00000 −0.183340
\(477\) 0 0
\(478\) 13.0000i 0.594606i
\(479\) 4.00000 0.182765 0.0913823 0.995816i \(-0.470871\pi\)
0.0913823 + 0.995816i \(0.470871\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) 2.00000i 0.0910975i
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 10.0000 20.0000i 0.454077 0.908153i
\(486\) 0 0
\(487\) 21.0000i 0.951601i −0.879553 0.475800i \(-0.842158\pi\)
0.879553 0.475800i \(-0.157842\pi\)
\(488\) 7.00000i 0.316875i
\(489\) 0 0
\(490\) 1.00000 2.00000i 0.0451754 0.0903508i
\(491\) −30.0000 −1.35388 −0.676941 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) 0 0
\(493\) 12.0000i 0.540453i
\(494\) 0 0
\(495\) 0 0
\(496\) 7.00000 0.314309
\(497\) 3.00000i 0.134568i
\(498\) 0 0
\(499\) −9.00000 −0.402895 −0.201448 0.979499i \(-0.564565\pi\)
−0.201448 + 0.979499i \(0.564565\pi\)
\(500\) 2.00000 + 11.0000i 0.0894427 + 0.491935i
\(501\) 0 0
\(502\) 12.0000i 0.535586i
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) −20.0000 10.0000i −0.889988 0.444994i
\(506\) −16.0000 −0.711287
\(507\) 0 0
\(508\) 7.00000i 0.310575i
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −7.00000 −0.309662
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 24.0000 1.05859
\(515\) 14.0000 28.0000i 0.616914 1.23383i
\(516\) 0 0
\(517\) 52.0000i 2.28696i
\(518\) 1.00000i 0.0439375i
\(519\) 0 0
\(520\) −12.0000 6.00000i −0.526235 0.263117i
\(521\) −38.0000 −1.66481 −0.832405 0.554168i \(-0.813037\pi\)
−0.832405 + 0.554168i \(0.813037\pi\)
\(522\) 0 0
\(523\) 28.0000i 1.22435i −0.790721 0.612177i \(-0.790294\pi\)
0.790721 0.612177i \(-0.209706\pi\)
\(524\) −15.0000 −0.655278
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) 28.0000i 1.21970i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) −12.0000 6.00000i −0.521247 0.260623i
\(531\) 0 0
\(532\) 0 0
\(533\) 42.0000i 1.81922i
\(534\) 0 0
\(535\) −9.00000 + 18.0000i −0.389104 + 0.778208i
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) 0 0
\(539\) 4.00000 0.172292
\(540\) 0 0
\(541\) −32.0000 −1.37579 −0.687894 0.725811i \(-0.741464\pi\)
−0.687894 + 0.725811i \(0.741464\pi\)
\(542\) 11.0000i 0.472490i
\(543\) 0 0
\(544\) −4.00000 −0.171499
\(545\) −40.0000 20.0000i −1.71341 0.856706i
\(546\) 0 0
\(547\) 32.0000i 1.36822i 0.729378 + 0.684111i \(0.239809\pi\)
−0.729378 + 0.684111i \(0.760191\pi\)
\(548\) 15.0000i 0.640768i
\(549\) 0 0
\(550\) −16.0000 + 12.0000i −0.682242 + 0.511682i
\(551\) 0 0
\(552\) 0 0
\(553\) 12.0000i 0.510292i
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −14.0000 −0.593732
\(557\) 12.0000i 0.508456i −0.967144 0.254228i \(-0.918179\pi\)
0.967144 0.254228i \(-0.0818214\pi\)
\(558\) 0 0
\(559\) 60.0000 2.53773
\(560\) 1.00000 2.00000i 0.0422577 0.0845154i
\(561\) 0 0
\(562\) 24.0000i 1.01238i
\(563\) 10.0000i 0.421450i −0.977545 0.210725i \(-0.932418\pi\)
0.977545 0.210725i \(-0.0675824\pi\)
\(564\) 0 0
\(565\) 3.00000 6.00000i 0.126211 0.252422i
\(566\) −7.00000 −0.294232
\(567\) 0 0
\(568\) 3.00000i 0.125877i
\(569\) −20.0000 −0.838444 −0.419222 0.907884i \(-0.637697\pi\)
−0.419222 + 0.907884i \(0.637697\pi\)
\(570\) 0 0
\(571\) 19.0000 0.795125 0.397563 0.917575i \(-0.369856\pi\)
0.397563 + 0.917575i \(0.369856\pi\)
\(572\) 24.0000i 1.00349i
\(573\) 0 0
\(574\) −7.00000 −0.292174
\(575\) −16.0000 + 12.0000i −0.667246 + 0.500435i
\(576\) 0 0
\(577\) 11.0000i 0.457936i 0.973434 + 0.228968i \(0.0735351\pi\)
−0.973434 + 0.228968i \(0.926465\pi\)
\(578\) 1.00000i 0.0415945i
\(579\) 0 0
\(580\) −6.00000 3.00000i −0.249136 0.124568i
\(581\) 2.00000 0.0829740
\(582\) 0 0
\(583\) 24.0000i 0.993978i
\(584\) −7.00000 −0.289662
\(585\) 0 0
\(586\) 15.0000 0.619644
\(587\) 18.0000i 0.742940i 0.928445 + 0.371470i \(0.121146\pi\)
−0.928445 + 0.371470i \(0.878854\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −5.00000 + 10.0000i −0.205847 + 0.411693i
\(591\) 0 0
\(592\) 1.00000i 0.0410997i
\(593\) 30.0000i 1.23195i 0.787765 + 0.615976i \(0.211238\pi\)
−0.787765 + 0.615976i \(0.788762\pi\)
\(594\) 0 0
\(595\) −8.00000 4.00000i −0.327968 0.163984i
\(596\) 10.0000 0.409616
\(597\) 0 0
\(598\) 24.0000i 0.981433i
\(599\) 15.0000 0.612883 0.306442 0.951889i \(-0.400862\pi\)
0.306442 + 0.951889i \(0.400862\pi\)
\(600\) 0 0
\(601\) 44.0000 1.79480 0.897399 0.441221i \(-0.145454\pi\)
0.897399 + 0.441221i \(0.145454\pi\)
\(602\) 10.0000i 0.407570i
\(603\) 0 0
\(604\) 14.0000 0.569652
\(605\) −10.0000 5.00000i −0.406558 0.203279i
\(606\) 0 0
\(607\) 14.0000i 0.568242i −0.958788 0.284121i \(-0.908298\pi\)
0.958788 0.284121i \(-0.0917018\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 7.00000 14.0000i 0.283422 0.566843i
\(611\) 78.0000 3.15554
\(612\) 0 0
\(613\) 22.0000i 0.888572i 0.895885 + 0.444286i \(0.146543\pi\)
−0.895885 + 0.444286i \(0.853457\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 4.00000 0.161165
\(617\) 31.0000i 1.24801i −0.781419 0.624007i \(-0.785504\pi\)
0.781419 0.624007i \(-0.214496\pi\)
\(618\) 0 0
\(619\) 40.0000 1.60774 0.803868 0.594808i \(-0.202772\pi\)
0.803868 + 0.594808i \(0.202772\pi\)
\(620\) 14.0000 + 7.00000i 0.562254 + 0.281127i
\(621\) 0 0
\(622\) 30.0000i 1.20289i
\(623\) 14.0000i 0.560898i
\(624\) 0 0
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) −26.0000 −1.03917
\(627\) 0 0
\(628\) 14.0000i 0.558661i
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 12.0000i 0.477334i
\(633\) 0 0
\(634\) 14.0000 0.556011
\(635\) −7.00000 + 14.0000i −0.277787 + 0.555573i
\(636\) 0 0
\(637\) 6.00000i 0.237729i
\(638\) 12.0000i 0.475085i
\(639\) 0 0
\(640\) 1.00000 2.00000i 0.0395285 0.0790569i
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 0 0
\(643\) 39.0000i 1.53801i −0.639243 0.769005i \(-0.720752\pi\)
0.639243 0.769005i \(-0.279248\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) 0 0
\(647\) 33.0000i 1.29736i 0.761060 + 0.648682i \(0.224679\pi\)
−0.761060 + 0.648682i \(0.775321\pi\)
\(648\) 0 0
\(649\) −20.0000 −0.785069
\(650\) −18.0000 24.0000i −0.706018 0.941357i
\(651\) 0 0
\(652\) 14.0000i 0.548282i
\(653\) 14.0000i 0.547862i −0.961749 0.273931i \(-0.911676\pi\)
0.961749 0.273931i \(-0.0883240\pi\)
\(654\) 0 0
\(655\) −30.0000 15.0000i −1.17220 0.586098i
\(656\) −7.00000 −0.273304
\(657\) 0 0
\(658\) 13.0000i 0.506793i
\(659\) 26.0000 1.01282 0.506408 0.862294i \(-0.330973\pi\)
0.506408 + 0.862294i \(0.330973\pi\)
\(660\) 0 0
\(661\) −19.0000 −0.739014 −0.369507 0.929228i \(-0.620473\pi\)
−0.369507 + 0.929228i \(0.620473\pi\)
\(662\) 35.0000i 1.36031i
\(663\) 0 0
\(664\) 2.00000 0.0776151
\(665\) 0 0
\(666\) 0 0
\(667\) 12.0000i 0.464642i
\(668\) 3.00000i 0.116073i
\(669\) 0 0
\(670\) −8.00000 4.00000i −0.309067 0.154533i
\(671\) 28.0000 1.08093
\(672\) 0 0
\(673\) 36.0000i 1.38770i 0.720121 + 0.693849i \(0.244086\pi\)
−0.720121 + 0.693849i \(0.755914\pi\)
\(674\) −16.0000 −0.616297
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) 29.0000i 1.11456i 0.830324 + 0.557280i \(0.188155\pi\)
−0.830324 + 0.557280i \(0.811845\pi\)
\(678\) 0 0
\(679\) −10.0000 −0.383765
\(680\) −8.00000 4.00000i −0.306786 0.153393i
\(681\) 0 0
\(682\) 28.0000i 1.07218i
\(683\) 1.00000i 0.0382639i 0.999817 + 0.0191320i \(0.00609027\pi\)
−0.999817 + 0.0191320i \(0.993910\pi\)
\(684\) 0 0
\(685\) −15.0000 + 30.0000i −0.573121 + 1.14624i
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 10.0000i 0.381246i
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) −38.0000 −1.44559 −0.722794 0.691063i \(-0.757142\pi\)
−0.722794 + 0.691063i \(0.757142\pi\)
\(692\) 17.0000i 0.646243i
\(693\) 0 0
\(694\) −27.0000 −1.02491
\(695\) −28.0000 14.0000i −1.06210 0.531050i
\(696\) 0 0
\(697\) 28.0000i 1.06058i
\(698\) 2.00000i 0.0757011i
\(699\) 0 0
\(700\) 4.00000 3.00000i 0.151186 0.113389i
\(701\) 35.0000 1.32193 0.660966 0.750416i \(-0.270147\pi\)
0.660966 + 0.750416i \(0.270147\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) 32.0000 1.20434
\(707\) 10.0000i 0.376089i
\(708\) 0 0
\(709\) −48.0000 −1.80268 −0.901339 0.433114i \(-0.857415\pi\)
−0.901339 + 0.433114i \(0.857415\pi\)
\(710\) 3.00000 6.00000i 0.112588 0.225176i
\(711\) 0 0
\(712\) 14.0000i 0.524672i
\(713\) 28.0000i 1.04861i
\(714\) 0 0
\(715\) 24.0000 48.0000i 0.897549 1.79510i
\(716\) 16.0000 0.597948
\(717\) 0 0
\(718\) 12.0000i 0.447836i
\(719\) −2.00000 −0.0745874 −0.0372937 0.999304i \(-0.511874\pi\)
−0.0372937 + 0.999304i \(0.511874\pi\)
\(720\) 0 0
\(721\) −14.0000 −0.521387
\(722\) 19.0000i 0.707107i
\(723\) 0 0
\(724\) −23.0000 −0.854788
\(725\) −9.00000 12.0000i −0.334252 0.445669i
\(726\) 0 0
\(727\) 6.00000i 0.222528i 0.993791 + 0.111264i \(0.0354899\pi\)
−0.993791 + 0.111264i \(0.964510\pi\)
\(728\) 6.00000i 0.222375i
\(729\) 0 0
\(730\) −14.0000 7.00000i −0.518163 0.259082i
\(731\) 40.0000 1.47945
\(732\) 0 0
\(733\) 44.0000i 1.62518i 0.582838 + 0.812589i \(0.301942\pi\)
−0.582838 + 0.812589i \(0.698058\pi\)
\(734\) 2.00000 0.0738213
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 16.0000i 0.589368i
\(738\) 0 0
\(739\) −13.0000 −0.478213 −0.239106 0.970993i \(-0.576854\pi\)
−0.239106 + 0.970993i \(0.576854\pi\)
\(740\) 1.00000 2.00000i 0.0367607 0.0735215i
\(741\) 0 0
\(742\) 6.00000i 0.220267i
\(743\) 18.0000i 0.660356i 0.943919 + 0.330178i \(0.107109\pi\)
−0.943919 + 0.330178i \(0.892891\pi\)
\(744\) 0 0
\(745\) 20.0000 + 10.0000i 0.732743 + 0.366372i
\(746\) −3.00000 −0.109838
\(747\) 0 0
\(748\) 16.0000i 0.585018i
\(749\) 9.00000 0.328853
\(750\) 0 0
\(751\) −22.0000 −0.802791 −0.401396 0.915905i \(-0.631475\pi\)
−0.401396 + 0.915905i \(0.631475\pi\)
\(752\) 13.0000i 0.474061i
\(753\) 0 0
\(754\) 18.0000 0.655521
\(755\) 28.0000 + 14.0000i 1.01902 + 0.509512i
\(756\) 0 0
\(757\) 19.0000i 0.690567i −0.938498 0.345283i \(-0.887783\pi\)
0.938498 0.345283i \(-0.112217\pi\)
\(758\) 1.00000i 0.0363216i
\(759\) 0 0
\(760\) 0 0
\(761\) 3.00000 0.108750 0.0543750 0.998521i \(-0.482683\pi\)
0.0543750 + 0.998521i \(0.482683\pi\)
\(762\) 0 0
\(763\) 20.0000i 0.724049i
\(764\) 0 0
\(765\) 0 0
\(766\) 8.00000 0.289052
\(767\) 30.0000i 1.08324i
\(768\) 0 0
\(769\) 12.0000 0.432731 0.216366 0.976312i \(-0.430580\pi\)
0.216366 + 0.976312i \(0.430580\pi\)
\(770\) 8.00000 + 4.00000i 0.288300 + 0.144150i
\(771\) 0 0
\(772\) 2.00000i 0.0719816i
\(773\) 27.0000i 0.971123i −0.874203 0.485561i \(-0.838615\pi\)
0.874203 0.485561i \(-0.161385\pi\)
\(774\) 0 0
\(775\) 21.0000 + 28.0000i 0.754342 + 1.00579i
\(776\) −10.0000 −0.358979
\(777\) 0 0
\(778\) 23.0000i 0.824590i
\(779\) 0 0
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 16.0000i 0.572159i
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) −14.0000 + 28.0000i −0.499681 + 0.999363i
\(786\) 0 0
\(787\) 7.00000i 0.249523i −0.992187 0.124762i \(-0.960183\pi\)
0.992187 0.124762i \(-0.0398166\pi\)
\(788\) 8.00000i 0.284988i
\(789\) 0 0
\(790\) 12.0000 24.0000i 0.426941 0.853882i
\(791\) −3.00000 −0.106668
\(792\) 0 0
\(793\) 42.0000i 1.49146i
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) 24.0000 0.850657
\(797\) 37.0000i 1.31061i −0.755366 0.655304i \(-0.772541\pi\)
0.755366 0.655304i \(-0.227459\pi\)
\(798\) 0 0
\(799\) 52.0000 1.83963
\(800\) 4.00000 3.00000i 0.141421 0.106066i
\(801\) 0 0
\(802\) 16.0000i 0.564980i
\(803\) 28.0000i 0.988099i
\(804\) 0 0
\(805\) 8.00000 + 4.00000i 0.281963 + 0.140981i
\(806\) −42.0000 −1.47939
\(807\) 0 0
\(808\) 10.0000i 0.351799i
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) −14.0000 −0.491606 −0.245803 0.969320i \(-0.579052\pi\)
−0.245803 + 0.969320i \(0.579052\pi\)
\(812\) 3.00000i 0.105279i
\(813\) 0 0
\(814\) 4.00000 0.140200
\(815\) 14.0000 28.0000i 0.490399 0.980797i
\(816\) 0 0
\(817\) 0 0
\(818\) 20.0000i 0.699284i
\(819\) 0 0
\(820\) −14.0000 7.00000i −0.488901 0.244451i
\(821\) −2.00000 −0.0698005 −0.0349002 0.999391i \(-0.511111\pi\)
−0.0349002 + 0.999391i \(0.511111\pi\)
\(822\) 0 0
\(823\) 51.0000i 1.77775i 0.458151 + 0.888874i \(0.348512\pi\)
−0.458151 + 0.888874i \(0.651488\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) 5.00000 0.173972
\(827\) 25.0000i 0.869335i −0.900591 0.434668i \(-0.856866\pi\)
0.900591 0.434668i \(-0.143134\pi\)
\(828\) 0 0
\(829\) −6.00000 −0.208389 −0.104194 0.994557i \(-0.533226\pi\)
−0.104194 + 0.994557i \(0.533226\pi\)
\(830\) 4.00000 + 2.00000i 0.138842 + 0.0694210i
\(831\) 0 0
\(832\) 6.00000i 0.208013i
\(833\) 4.00000i 0.138592i
\(834\) 0 0
\(835\) −3.00000 + 6.00000i −0.103819 + 0.207639i
\(836\) 0 0
\(837\) 0 0
\(838\) 25.0000i 0.863611i
\(839\) −4.00000 −0.138095 −0.0690477 0.997613i \(-0.521996\pi\)
−0.0690477 + 0.997613i \(0.521996\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 20.0000i 0.689246i
\(843\) 0 0
\(844\) 15.0000 0.516321
\(845\) 46.0000 + 23.0000i 1.58245 + 0.791224i
\(846\) 0 0
\(847\) 5.00000i 0.171802i
\(848\) 6.00000i 0.206041i
\(849\) 0 0
\(850\) −12.0000 16.0000i −0.411597 0.548795i
\(851\) 4.00000 0.137118
\(852\) 0 0
\(853\) 16.0000i 0.547830i −0.961754 0.273915i \(-0.911681\pi\)
0.961754 0.273915i \(-0.0883186\pi\)
\(854\) −7.00000 −0.239535
\(855\) 0 0
\(856\) 9.00000 0.307614
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) −10.0000 + 20.0000i −0.340997 + 0.681994i
\(861\) 0 0
\(862\) 11.0000i 0.374661i
\(863\) 6.00000i 0.204242i 0.994772 + 0.102121i \(0.0325630\pi\)
−0.994772 + 0.102121i \(0.967437\pi\)
\(864\) 0 0
\(865\) 17.0000 34.0000i 0.578017 1.15603i
\(866\) −29.0000 −0.985460
\(867\) 0 0
\(868\) 7.00000i 0.237595i
\(869\) 48.0000 1.62829
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) 20.0000i 0.677285i
\(873\) 0 0
\(874\) 0 0
\(875\) 11.0000 2.00000i 0.371868 0.0676123i
\(876\) 0 0
\(877\) 49.0000i 1.65461i 0.561751 + 0.827306i \(0.310128\pi\)
−0.561751 + 0.827306i \(0.689872\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 8.00000 + 4.00000i 0.269680 + 0.134840i
\(881\) 33.0000 1.11180 0.555899 0.831250i \(-0.312374\pi\)
0.555899 + 0.831250i \(0.312374\pi\)
\(882\) 0 0
\(883\) 54.0000i 1.81724i 0.417619 + 0.908622i \(0.362865\pi\)
−0.417619 + 0.908622i \(0.637135\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) 9.00000 0.302361
\(887\) 39.0000i 1.30949i −0.755849 0.654746i \(-0.772776\pi\)
0.755849 0.654746i \(-0.227224\pi\)
\(888\) 0 0
\(889\) 7.00000 0.234772
\(890\) 14.0000 28.0000i 0.469281 0.938562i
\(891\) 0 0
\(892\) 4.00000i 0.133930i
\(893\) 0 0
\(894\) 0 0
\(895\) 32.0000 + 16.0000i 1.06964 + 0.534821i
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 36.0000i 1.20134i
\(899\) −21.0000 −0.700389
\(900\) 0 0
\(901\) 24.0000 0.799556
\(902\) 28.0000i 0.932298i
\(903\) 0 0
\(904\) −3.00000 −0.0997785
\(905\) −46.0000 23.0000i −1.52909 0.764546i
\(906\) 0 0
\(907\) 24.0000i 0.796907i −0.917189 0.398453i \(-0.869547\pi\)
0.917189 0.398453i \(-0.130453\pi\)
\(908\) 16.0000i 0.530979i
\(909\) 0 0
\(910\) −6.00000 + 12.0000i −0.198898 + 0.397796i
\(911\) 9.00000 0.298183 0.149092 0.988823i \(-0.452365\pi\)
0.149092 + 0.988823i \(0.452365\pi\)
\(912\) 0 0
\(913\) 8.00000i 0.264761i
\(914\) 8.00000 0.264616
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) 15.0000i 0.495344i
\(918\) 0 0
\(919\) −18.0000 −0.593765 −0.296883 0.954914i \(-0.595947\pi\)
−0.296883 + 0.954914i \(0.595947\pi\)
\(920\) 8.00000 + 4.00000i 0.263752 + 0.131876i
\(921\) 0 0
\(922\) 36.0000i 1.18560i
\(923\) 18.0000i 0.592477i
\(924\) 0 0
\(925\) 4.00000 3.00000i 0.131519 0.0986394i
\(926\) 33.0000 1.08445
\(927\) 0 0
\(928\) 3.00000i 0.0984798i
\(929\) 42.0000 1.37798 0.688988 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 11.0000i 0.360317i
\(933\) 0 0
\(934\) 18.0000 0.588978
\(935\) 16.0000 32.0000i 0.523256 1.04651i
\(936\) 0 0
\(937\) 23.0000i 0.751377i −0.926746 0.375689i \(-0.877406\pi\)
0.926746 0.375689i \(-0.122594\pi\)
\(938\) 4.00000i 0.130605i
\(939\) 0 0
\(940\) −13.0000 + 26.0000i −0.424013 + 0.848026i
\(941\) −40.0000 −1.30396 −0.651981 0.758235i \(-0.726062\pi\)
−0.651981 + 0.758235i \(0.726062\pi\)
\(942\) 0 0
\(943\) 28.0000i 0.911805i
\(944\) 5.00000 0.162736
\(945\) 0 0
\(946\) −40.0000 −1.30051
\(947\) 12.0000i 0.389948i −0.980808 0.194974i \(-0.937538\pi\)
0.980808 0.194974i \(-0.0624622\pi\)
\(948\) 0 0
\(949\) 42.0000 1.36338
\(950\) 0 0
\(951\) 0 0
\(952\) 4.00000i 0.129641i
\(953\) 38.0000i 1.23094i 0.788160 + 0.615470i \(0.211034\pi\)
−0.788160 + 0.615470i \(0.788966\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −13.0000 −0.420450
\(957\) 0 0
\(958\) 4.00000i 0.129234i
\(959\) 15.0000 0.484375
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 6.00000i 0.193448i
\(963\) 0 0
\(964\) 2.00000 0.0644157
\(965\) −2.00000 + 4.00000i −0.0643823 + 0.128765i
\(966\) 0 0
\(967\) 29.0000i 0.932577i −0.884633 0.466289i \(-0.845591\pi\)
0.884633 0.466289i \(-0.154409\pi\)
\(968\) 5.00000i 0.160706i
\(969\) 0 0
\(970\) −20.0000 10.0000i −0.642161 0.321081i
\(971\) 33.0000 1.05902 0.529510 0.848304i \(-0.322376\pi\)
0.529510 + 0.848304i \(0.322376\pi\)
\(972\) 0 0
\(973\) 14.0000i 0.448819i
\(974\) −21.0000 −0.672883
\(975\) 0 0
\(976\) −7.00000 −0.224065
\(977\) 18.0000i 0.575871i 0.957650 + 0.287936i \(0.0929689\pi\)
−0.957650 + 0.287936i \(0.907031\pi\)
\(978\) 0 0
\(979\) 56.0000 1.78977
\(980\) −2.00000 1.00000i −0.0638877 0.0319438i
\(981\) 0 0
\(982\) 30.0000i 0.957338i
\(983\) 52.0000i 1.65854i −0.558846 0.829271i \(-0.688756\pi\)
0.558846 0.829271i \(-0.311244\pi\)
\(984\) 0 0
\(985\) −8.00000 + 16.0000i −0.254901 + 0.509802i
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 0 0
\(989\) −40.0000 −1.27193
\(990\) 0 0
\(991\) −2.00000 −0.0635321 −0.0317660 0.999495i \(-0.510113\pi\)
−0.0317660 + 0.999495i \(0.510113\pi\)
\(992\) 7.00000i 0.222250i
\(993\) 0 0
\(994\) −3.00000 −0.0951542
\(995\) 48.0000 + 24.0000i 1.52170 + 0.760851i
\(996\) 0 0
\(997\) 12.0000i 0.380044i −0.981780 0.190022i \(-0.939144\pi\)
0.981780 0.190022i \(-0.0608559\pi\)
\(998\) 9.00000i 0.284890i
\(999\) 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 1890.2.g.a.379.1 2
3.2 odd 2 1890.2.g.k.379.2 yes 2
5.2 odd 4 9450.2.a.cc.1.1 1
5.3 odd 4 9450.2.a.be.1.1 1
5.4 even 2 inner 1890.2.g.a.379.2 yes 2
15.2 even 4 9450.2.a.w.1.1 1
15.8 even 4 9450.2.a.dx.1.1 1
15.14 odd 2 1890.2.g.k.379.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1890.2.g.a.379.1 2 1.1 even 1 trivial
1890.2.g.a.379.2 yes 2 5.4 even 2 inner
1890.2.g.k.379.1 yes 2 15.14 odd 2
1890.2.g.k.379.2 yes 2 3.2 odd 2
9450.2.a.w.1.1 1 15.2 even 4
9450.2.a.be.1.1 1 5.3 odd 4
9450.2.a.cc.1.1 1 5.2 odd 4
9450.2.a.dx.1.1 1 15.8 even 4