Properties

Label 9225.2.a.bx.1.2
Level $9225$
Weight $2$
Character 9225.1
Self dual yes
Analytic conductor $73.662$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9225,2,Mod(1,9225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9225 = 3^{2} \cdot 5^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9225.a (trivial)

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: yes
Analytic conductor: \(73.6619958646\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 123)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.470683\) of defining polynomial
Character \(\chi\) \(=\) 9225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.470683 q^{2} -1.77846 q^{4} -3.30777 q^{7} -1.77846 q^{8} +O(q^{10})\) \(q+0.470683 q^{2} -1.77846 q^{4} -3.30777 q^{7} -1.77846 q^{8} +1.47068 q^{11} +0.249141 q^{13} -1.55691 q^{14} +2.71982 q^{16} -5.02760 q^{17} -2.24914 q^{19} +0.692226 q^{22} -6.24914 q^{23} +0.117266 q^{26} +5.88273 q^{28} +2.41205 q^{29} -2.89572 q^{31} +4.83709 q^{32} -2.36641 q^{34} -9.71982 q^{37} -1.05863 q^{38} -1.00000 q^{41} +10.8337 q^{43} -2.61555 q^{44} -2.94137 q^{46} -4.08623 q^{47} +3.94137 q^{49} -0.443086 q^{52} -1.43965 q^{53} +5.88273 q^{56} +1.13531 q^{58} -2.61555 q^{59} -5.71982 q^{61} -1.36297 q^{62} -3.16291 q^{64} -15.9379 q^{67} +8.94137 q^{68} +10.5293 q^{71} -9.39400 q^{73} -4.57496 q^{74} +4.00000 q^{76} -4.86469 q^{77} -0.560352 q^{79} -0.470683 q^{82} +3.80605 q^{83} +5.09922 q^{86} -2.61555 q^{88} -4.11727 q^{89} -0.824101 q^{91} +11.1138 q^{92} -1.92332 q^{94} +7.19051 q^{97} +1.85514 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8} + 4 q^{11} - 8 q^{13} + 12 q^{14} - q^{16} + 2 q^{17} + 2 q^{19} + 10 q^{22} - 10 q^{23} + 2 q^{26} + 16 q^{28} + 6 q^{29} - 2 q^{31} + 7 q^{32} - 20 q^{37} - 4 q^{38} - 3 q^{41} - 10 q^{43} + 8 q^{44} - 8 q^{46} + 4 q^{47} + 11 q^{49} - 18 q^{52} + 14 q^{53} + 16 q^{56} + 28 q^{58} + 8 q^{59} - 8 q^{61} + 38 q^{62} - 17 q^{64} - 12 q^{67} + 26 q^{68} + 32 q^{71} - 4 q^{73} - 20 q^{74} + 12 q^{76} + 10 q^{77} - 20 q^{79} - q^{82} - 14 q^{83} - 6 q^{86} + 8 q^{88} - 14 q^{89} + 18 q^{94} + 12 q^{97} + 21 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.470683 0.332823 0.166412 0.986056i \(-0.446782\pi\)
0.166412 + 0.986056i \(0.446782\pi\)
\(3\) 0 0
\(4\) −1.77846 −0.889229
\(5\) 0 0
\(6\) 0 0
\(7\) −3.30777 −1.25022 −0.625110 0.780536i \(-0.714946\pi\)
−0.625110 + 0.780536i \(0.714946\pi\)
\(8\) −1.77846 −0.628780
\(9\) 0 0
\(10\) 0 0
\(11\) 1.47068 0.443428 0.221714 0.975112i \(-0.428835\pi\)
0.221714 + 0.975112i \(0.428835\pi\)
\(12\) 0 0
\(13\) 0.249141 0.0690992 0.0345496 0.999403i \(-0.489000\pi\)
0.0345496 + 0.999403i \(0.489000\pi\)
\(14\) −1.55691 −0.416103
\(15\) 0 0
\(16\) 2.71982 0.679956
\(17\) −5.02760 −1.21937 −0.609686 0.792643i \(-0.708704\pi\)
−0.609686 + 0.792643i \(0.708704\pi\)
\(18\) 0 0
\(19\) −2.24914 −0.515988 −0.257994 0.966146i \(-0.583062\pi\)
−0.257994 + 0.966146i \(0.583062\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.692226 0.147583
\(23\) −6.24914 −1.30304 −0.651518 0.758633i \(-0.725867\pi\)
−0.651518 + 0.758633i \(0.725867\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.117266 0.0229978
\(27\) 0 0
\(28\) 5.88273 1.11173
\(29\) 2.41205 0.447906 0.223953 0.974600i \(-0.428104\pi\)
0.223953 + 0.974600i \(0.428104\pi\)
\(30\) 0 0
\(31\) −2.89572 −0.520087 −0.260044 0.965597i \(-0.583737\pi\)
−0.260044 + 0.965597i \(0.583737\pi\)
\(32\) 4.83709 0.855085
\(33\) 0 0
\(34\) −2.36641 −0.405835
\(35\) 0 0
\(36\) 0 0
\(37\) −9.71982 −1.59793 −0.798965 0.601378i \(-0.794619\pi\)
−0.798965 + 0.601378i \(0.794619\pi\)
\(38\) −1.05863 −0.171733
\(39\) 0 0
\(40\) 0 0
\(41\) −1.00000 −0.156174
\(42\) 0 0
\(43\) 10.8337 1.65212 0.826058 0.563585i \(-0.190578\pi\)
0.826058 + 0.563585i \(0.190578\pi\)
\(44\) −2.61555 −0.394309
\(45\) 0 0
\(46\) −2.94137 −0.433681
\(47\) −4.08623 −0.596038 −0.298019 0.954560i \(-0.596326\pi\)
−0.298019 + 0.954560i \(0.596326\pi\)
\(48\) 0 0
\(49\) 3.94137 0.563052
\(50\) 0 0
\(51\) 0 0
\(52\) −0.443086 −0.0614449
\(53\) −1.43965 −0.197751 −0.0988754 0.995100i \(-0.531525\pi\)
−0.0988754 + 0.995100i \(0.531525\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 5.88273 0.786113
\(57\) 0 0
\(58\) 1.13531 0.149074
\(59\) −2.61555 −0.340515 −0.170258 0.985400i \(-0.554460\pi\)
−0.170258 + 0.985400i \(0.554460\pi\)
\(60\) 0 0
\(61\) −5.71982 −0.732348 −0.366174 0.930546i \(-0.619333\pi\)
−0.366174 + 0.930546i \(0.619333\pi\)
\(62\) −1.36297 −0.173097
\(63\) 0 0
\(64\) −3.16291 −0.395364
\(65\) 0 0
\(66\) 0 0
\(67\) −15.9379 −1.94713 −0.973564 0.228415i \(-0.926646\pi\)
−0.973564 + 0.228415i \(0.926646\pi\)
\(68\) 8.94137 1.08430
\(69\) 0 0
\(70\) 0 0
\(71\) 10.5293 1.24960 0.624800 0.780785i \(-0.285181\pi\)
0.624800 + 0.780785i \(0.285181\pi\)
\(72\) 0 0
\(73\) −9.39400 −1.09949 −0.549743 0.835334i \(-0.685274\pi\)
−0.549743 + 0.835334i \(0.685274\pi\)
\(74\) −4.57496 −0.531828
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) −4.86469 −0.554383
\(78\) 0 0
\(79\) −0.560352 −0.0630445 −0.0315223 0.999503i \(-0.510036\pi\)
−0.0315223 + 0.999503i \(0.510036\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −0.470683 −0.0519783
\(83\) 3.80605 0.417769 0.208884 0.977940i \(-0.433017\pi\)
0.208884 + 0.977940i \(0.433017\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 5.09922 0.549863
\(87\) 0 0
\(88\) −2.61555 −0.278818
\(89\) −4.11727 −0.436429 −0.218215 0.975901i \(-0.570023\pi\)
−0.218215 + 0.975901i \(0.570023\pi\)
\(90\) 0 0
\(91\) −0.824101 −0.0863892
\(92\) 11.1138 1.15870
\(93\) 0 0
\(94\) −1.92332 −0.198375
\(95\) 0 0
\(96\) 0 0
\(97\) 7.19051 0.730085 0.365043 0.930991i \(-0.381054\pi\)
0.365043 + 0.930991i \(0.381054\pi\)
\(98\) 1.85514 0.187397
\(99\) 0 0
\(100\) 0 0
\(101\) 11.1449 1.10896 0.554478 0.832199i \(-0.312918\pi\)
0.554478 + 0.832199i \(0.312918\pi\)
\(102\) 0 0
\(103\) −13.2767 −1.30820 −0.654098 0.756410i \(-0.726952\pi\)
−0.654098 + 0.756410i \(0.726952\pi\)
\(104\) −0.443086 −0.0434481
\(105\) 0 0
\(106\) −0.677618 −0.0658161
\(107\) 9.32238 0.901229 0.450614 0.892719i \(-0.351205\pi\)
0.450614 + 0.892719i \(0.351205\pi\)
\(108\) 0 0
\(109\) −0.249141 −0.0238633 −0.0119317 0.999929i \(-0.503798\pi\)
−0.0119317 + 0.999929i \(0.503798\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −8.99656 −0.850095
\(113\) 9.24570 0.869763 0.434881 0.900488i \(-0.356790\pi\)
0.434881 + 0.900488i \(0.356790\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −4.28973 −0.398291
\(117\) 0 0
\(118\) −1.23109 −0.113331
\(119\) 16.6302 1.52448
\(120\) 0 0
\(121\) −8.83709 −0.803372
\(122\) −2.69223 −0.243743
\(123\) 0 0
\(124\) 5.14992 0.462476
\(125\) 0 0
\(126\) 0 0
\(127\) 0.824101 0.0731271 0.0365635 0.999331i \(-0.488359\pi\)
0.0365635 + 0.999331i \(0.488359\pi\)
\(128\) −11.1629 −0.986671
\(129\) 0 0
\(130\) 0 0
\(131\) −8.13187 −0.710485 −0.355243 0.934774i \(-0.615602\pi\)
−0.355243 + 0.934774i \(0.615602\pi\)
\(132\) 0 0
\(133\) 7.43965 0.645099
\(134\) −7.50172 −0.648050
\(135\) 0 0
\(136\) 8.94137 0.766716
\(137\) −12.2035 −1.04262 −0.521308 0.853369i \(-0.674556\pi\)
−0.521308 + 0.853369i \(0.674556\pi\)
\(138\) 0 0
\(139\) 14.2897 1.21204 0.606019 0.795450i \(-0.292765\pi\)
0.606019 + 0.795450i \(0.292765\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 4.95597 0.415896
\(143\) 0.366407 0.0306405
\(144\) 0 0
\(145\) 0 0
\(146\) −4.42160 −0.365934
\(147\) 0 0
\(148\) 17.2863 1.42092
\(149\) 22.1104 1.81135 0.905677 0.423969i \(-0.139363\pi\)
0.905677 + 0.423969i \(0.139363\pi\)
\(150\) 0 0
\(151\) 10.1173 0.823331 0.411666 0.911335i \(-0.364947\pi\)
0.411666 + 0.911335i \(0.364947\pi\)
\(152\) 4.00000 0.324443
\(153\) 0 0
\(154\) −2.28973 −0.184512
\(155\) 0 0
\(156\) 0 0
\(157\) −18.7620 −1.49737 −0.748686 0.662924i \(-0.769315\pi\)
−0.748686 + 0.662924i \(0.769315\pi\)
\(158\) −0.263748 −0.0209827
\(159\) 0 0
\(160\) 0 0
\(161\) 20.6707 1.62908
\(162\) 0 0
\(163\) −8.54392 −0.669212 −0.334606 0.942358i \(-0.608603\pi\)
−0.334606 + 0.942358i \(0.608603\pi\)
\(164\) 1.77846 0.138874
\(165\) 0 0
\(166\) 1.79145 0.139043
\(167\) 5.05863 0.391449 0.195724 0.980659i \(-0.437294\pi\)
0.195724 + 0.980659i \(0.437294\pi\)
\(168\) 0 0
\(169\) −12.9379 −0.995225
\(170\) 0 0
\(171\) 0 0
\(172\) −19.2672 −1.46911
\(173\) 18.8647 1.43426 0.717128 0.696942i \(-0.245456\pi\)
0.717128 + 0.696942i \(0.245456\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) −1.93793 −0.145254
\(179\) 2.85514 0.213403 0.106701 0.994291i \(-0.465971\pi\)
0.106701 + 0.994291i \(0.465971\pi\)
\(180\) 0 0
\(181\) 13.4396 0.998961 0.499481 0.866325i \(-0.333524\pi\)
0.499481 + 0.866325i \(0.333524\pi\)
\(182\) −0.387890 −0.0287524
\(183\) 0 0
\(184\) 11.1138 0.819322
\(185\) 0 0
\(186\) 0 0
\(187\) −7.39400 −0.540703
\(188\) 7.26719 0.530014
\(189\) 0 0
\(190\) 0 0
\(191\) −13.4948 −0.976453 −0.488226 0.872717i \(-0.662356\pi\)
−0.488226 + 0.872717i \(0.662356\pi\)
\(192\) 0 0
\(193\) 8.01461 0.576904 0.288452 0.957494i \(-0.406859\pi\)
0.288452 + 0.957494i \(0.406859\pi\)
\(194\) 3.38445 0.242990
\(195\) 0 0
\(196\) −7.00955 −0.500682
\(197\) 13.9785 0.995928 0.497964 0.867198i \(-0.334081\pi\)
0.497964 + 0.867198i \(0.334081\pi\)
\(198\) 0 0
\(199\) −4.86469 −0.344849 −0.172424 0.985023i \(-0.555160\pi\)
−0.172424 + 0.985023i \(0.555160\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 5.24570 0.369086
\(203\) −7.97852 −0.559982
\(204\) 0 0
\(205\) 0 0
\(206\) −6.24914 −0.435398
\(207\) 0 0
\(208\) 0.677618 0.0469844
\(209\) −3.30777 −0.228803
\(210\) 0 0
\(211\) −10.6155 −0.730804 −0.365402 0.930850i \(-0.619069\pi\)
−0.365402 + 0.930850i \(0.619069\pi\)
\(212\) 2.56035 0.175846
\(213\) 0 0
\(214\) 4.38789 0.299950
\(215\) 0 0
\(216\) 0 0
\(217\) 9.57840 0.650224
\(218\) −0.117266 −0.00794228
\(219\) 0 0
\(220\) 0 0
\(221\) −1.25258 −0.0842575
\(222\) 0 0
\(223\) 17.9931 1.20491 0.602454 0.798153i \(-0.294190\pi\)
0.602454 + 0.798153i \(0.294190\pi\)
\(224\) −16.0000 −1.06904
\(225\) 0 0
\(226\) 4.35180 0.289477
\(227\) −0.910331 −0.0604208 −0.0302104 0.999544i \(-0.509618\pi\)
−0.0302104 + 0.999544i \(0.509618\pi\)
\(228\) 0 0
\(229\) −7.42504 −0.490660 −0.245330 0.969440i \(-0.578896\pi\)
−0.245330 + 0.969440i \(0.578896\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −4.28973 −0.281634
\(233\) 14.7620 0.967093 0.483546 0.875319i \(-0.339348\pi\)
0.483546 + 0.875319i \(0.339348\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 4.65164 0.302796
\(237\) 0 0
\(238\) 7.82754 0.507384
\(239\) 1.38445 0.0895528 0.0447764 0.998997i \(-0.485742\pi\)
0.0447764 + 0.998997i \(0.485742\pi\)
\(240\) 0 0
\(241\) −5.95436 −0.383554 −0.191777 0.981439i \(-0.561425\pi\)
−0.191777 + 0.981439i \(0.561425\pi\)
\(242\) −4.15947 −0.267381
\(243\) 0 0
\(244\) 10.1725 0.651225
\(245\) 0 0
\(246\) 0 0
\(247\) −0.560352 −0.0356543
\(248\) 5.14992 0.327020
\(249\) 0 0
\(250\) 0 0
\(251\) −2.70683 −0.170854 −0.0854269 0.996344i \(-0.527225\pi\)
−0.0854269 + 0.996344i \(0.527225\pi\)
\(252\) 0 0
\(253\) −9.19051 −0.577802
\(254\) 0.387890 0.0243384
\(255\) 0 0
\(256\) 1.07162 0.0669764
\(257\) −0.793065 −0.0494700 −0.0247350 0.999694i \(-0.507874\pi\)
−0.0247350 + 0.999694i \(0.507874\pi\)
\(258\) 0 0
\(259\) 32.1510 1.99776
\(260\) 0 0
\(261\) 0 0
\(262\) −3.82754 −0.236466
\(263\) −15.5879 −0.961194 −0.480597 0.876942i \(-0.659580\pi\)
−0.480597 + 0.876942i \(0.659580\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 3.50172 0.214704
\(267\) 0 0
\(268\) 28.3449 1.73144
\(269\) −3.05863 −0.186488 −0.0932441 0.995643i \(-0.529724\pi\)
−0.0932441 + 0.995643i \(0.529724\pi\)
\(270\) 0 0
\(271\) 12.2802 0.745968 0.372984 0.927838i \(-0.378335\pi\)
0.372984 + 0.927838i \(0.378335\pi\)
\(272\) −13.6742 −0.829119
\(273\) 0 0
\(274\) −5.74398 −0.347007
\(275\) 0 0
\(276\) 0 0
\(277\) 8.56990 0.514916 0.257458 0.966290i \(-0.417115\pi\)
0.257458 + 0.966290i \(0.417115\pi\)
\(278\) 6.72594 0.403395
\(279\) 0 0
\(280\) 0 0
\(281\) 14.5845 0.870039 0.435020 0.900421i \(-0.356741\pi\)
0.435020 + 0.900421i \(0.356741\pi\)
\(282\) 0 0
\(283\) 20.9509 1.24540 0.622701 0.782460i \(-0.286035\pi\)
0.622701 + 0.782460i \(0.286035\pi\)
\(284\) −18.7259 −1.11118
\(285\) 0 0
\(286\) 0.172462 0.0101979
\(287\) 3.30777 0.195252
\(288\) 0 0
\(289\) 8.27674 0.486867
\(290\) 0 0
\(291\) 0 0
\(292\) 16.7068 0.977694
\(293\) 31.4086 1.83491 0.917455 0.397839i \(-0.130240\pi\)
0.917455 + 0.397839i \(0.130240\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 17.2863 1.00475
\(297\) 0 0
\(298\) 10.4070 0.602861
\(299\) −1.55691 −0.0900387
\(300\) 0 0
\(301\) −35.8353 −2.06551
\(302\) 4.76203 0.274024
\(303\) 0 0
\(304\) −6.11727 −0.350849
\(305\) 0 0
\(306\) 0 0
\(307\) −8.54392 −0.487628 −0.243814 0.969822i \(-0.578399\pi\)
−0.243814 + 0.969822i \(0.578399\pi\)
\(308\) 8.65164 0.492973
\(309\) 0 0
\(310\) 0 0
\(311\) 5.49484 0.311584 0.155792 0.987790i \(-0.450207\pi\)
0.155792 + 0.987790i \(0.450207\pi\)
\(312\) 0 0
\(313\) −5.48024 −0.309761 −0.154881 0.987933i \(-0.549499\pi\)
−0.154881 + 0.987933i \(0.549499\pi\)
\(314\) −8.83098 −0.498361
\(315\) 0 0
\(316\) 0.996562 0.0560610
\(317\) −11.3173 −0.635644 −0.317822 0.948150i \(-0.602951\pi\)
−0.317822 + 0.948150i \(0.602951\pi\)
\(318\) 0 0
\(319\) 3.54736 0.198614
\(320\) 0 0
\(321\) 0 0
\(322\) 9.72938 0.542197
\(323\) 11.3078 0.629181
\(324\) 0 0
\(325\) 0 0
\(326\) −4.02148 −0.222729
\(327\) 0 0
\(328\) 1.77846 0.0981989
\(329\) 13.5163 0.745179
\(330\) 0 0
\(331\) 33.8613 1.86118 0.930591 0.366060i \(-0.119293\pi\)
0.930591 + 0.366060i \(0.119293\pi\)
\(332\) −6.76891 −0.371492
\(333\) 0 0
\(334\) 2.38101 0.130283
\(335\) 0 0
\(336\) 0 0
\(337\) 6.77846 0.369246 0.184623 0.982809i \(-0.440894\pi\)
0.184623 + 0.982809i \(0.440894\pi\)
\(338\) −6.08967 −0.331234
\(339\) 0 0
\(340\) 0 0
\(341\) −4.25869 −0.230621
\(342\) 0 0
\(343\) 10.1173 0.546281
\(344\) −19.2672 −1.03882
\(345\) 0 0
\(346\) 8.87930 0.477354
\(347\) 19.2001 1.03071 0.515357 0.856976i \(-0.327659\pi\)
0.515357 + 0.856976i \(0.327659\pi\)
\(348\) 0 0
\(349\) −6.54392 −0.350288 −0.175144 0.984543i \(-0.556039\pi\)
−0.175144 + 0.984543i \(0.556039\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 7.11383 0.379168
\(353\) −15.9931 −0.851228 −0.425614 0.904905i \(-0.639942\pi\)
−0.425614 + 0.904905i \(0.639942\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 7.32238 0.388085
\(357\) 0 0
\(358\) 1.34387 0.0710255
\(359\) 12.1319 0.640296 0.320148 0.947368i \(-0.396267\pi\)
0.320148 + 0.947368i \(0.396267\pi\)
\(360\) 0 0
\(361\) −13.9414 −0.733756
\(362\) 6.32582 0.332478
\(363\) 0 0
\(364\) 1.46563 0.0768198
\(365\) 0 0
\(366\) 0 0
\(367\) −6.39744 −0.333944 −0.166972 0.985962i \(-0.553399\pi\)
−0.166972 + 0.985962i \(0.553399\pi\)
\(368\) −16.9966 −0.886007
\(369\) 0 0
\(370\) 0 0
\(371\) 4.76203 0.247232
\(372\) 0 0
\(373\) 7.54049 0.390432 0.195216 0.980760i \(-0.437459\pi\)
0.195216 + 0.980760i \(0.437459\pi\)
\(374\) −3.48024 −0.179959
\(375\) 0 0
\(376\) 7.26719 0.374777
\(377\) 0.600939 0.0309500
\(378\) 0 0
\(379\) −0.0620710 −0.00318837 −0.00159419 0.999999i \(-0.500507\pi\)
−0.00159419 + 0.999999i \(0.500507\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −6.35180 −0.324986
\(383\) 23.9329 1.22291 0.611456 0.791278i \(-0.290584\pi\)
0.611456 + 0.791278i \(0.290584\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 3.77234 0.192007
\(387\) 0 0
\(388\) −12.7880 −0.649213
\(389\) 23.8613 1.20981 0.604907 0.796296i \(-0.293210\pi\)
0.604907 + 0.796296i \(0.293210\pi\)
\(390\) 0 0
\(391\) 31.4182 1.58888
\(392\) −7.00955 −0.354036
\(393\) 0 0
\(394\) 6.57946 0.331468
\(395\) 0 0
\(396\) 0 0
\(397\) −14.6302 −0.734266 −0.367133 0.930168i \(-0.619661\pi\)
−0.367133 + 0.930168i \(0.619661\pi\)
\(398\) −2.28973 −0.114774
\(399\) 0 0
\(400\) 0 0
\(401\) 11.3224 0.565413 0.282706 0.959206i \(-0.408768\pi\)
0.282706 + 0.959206i \(0.408768\pi\)
\(402\) 0 0
\(403\) −0.721442 −0.0359376
\(404\) −19.8207 −0.986115
\(405\) 0 0
\(406\) −3.75536 −0.186375
\(407\) −14.2948 −0.708566
\(408\) 0 0
\(409\) −33.8923 −1.67587 −0.837933 0.545773i \(-0.816236\pi\)
−0.837933 + 0.545773i \(0.816236\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 23.6121 1.16329
\(413\) 8.65164 0.425719
\(414\) 0 0
\(415\) 0 0
\(416\) 1.20512 0.0590856
\(417\) 0 0
\(418\) −1.55691 −0.0761512
\(419\) 1.98539 0.0969928 0.0484964 0.998823i \(-0.484557\pi\)
0.0484964 + 0.998823i \(0.484557\pi\)
\(420\) 0 0
\(421\) −5.67418 −0.276543 −0.138271 0.990394i \(-0.544155\pi\)
−0.138271 + 0.990394i \(0.544155\pi\)
\(422\) −4.99656 −0.243229
\(423\) 0 0
\(424\) 2.56035 0.124342
\(425\) 0 0
\(426\) 0 0
\(427\) 18.9199 0.915597
\(428\) −16.5795 −0.801398
\(429\) 0 0
\(430\) 0 0
\(431\) −1.75086 −0.0843359 −0.0421680 0.999111i \(-0.513426\pi\)
−0.0421680 + 0.999111i \(0.513426\pi\)
\(432\) 0 0
\(433\) 34.0647 1.63705 0.818524 0.574473i \(-0.194793\pi\)
0.818524 + 0.574473i \(0.194793\pi\)
\(434\) 4.50839 0.216410
\(435\) 0 0
\(436\) 0.443086 0.0212200
\(437\) 14.0552 0.672351
\(438\) 0 0
\(439\) −4.32582 −0.206460 −0.103230 0.994658i \(-0.532918\pi\)
−0.103230 + 0.994658i \(0.532918\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −0.589568 −0.0280429
\(443\) 36.9966 1.75776 0.878880 0.477043i \(-0.158291\pi\)
0.878880 + 0.477043i \(0.158291\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 8.46907 0.401022
\(447\) 0 0
\(448\) 10.4622 0.494292
\(449\) 15.9448 0.752482 0.376241 0.926522i \(-0.377216\pi\)
0.376241 + 0.926522i \(0.377216\pi\)
\(450\) 0 0
\(451\) −1.47068 −0.0692518
\(452\) −16.4431 −0.773418
\(453\) 0 0
\(454\) −0.428478 −0.0201095
\(455\) 0 0
\(456\) 0 0
\(457\) 29.4396 1.37713 0.688564 0.725175i \(-0.258241\pi\)
0.688564 + 0.725175i \(0.258241\pi\)
\(458\) −3.49484 −0.163303
\(459\) 0 0
\(460\) 0 0
\(461\) −21.8759 −1.01886 −0.509430 0.860512i \(-0.670144\pi\)
−0.509430 + 0.860512i \(0.670144\pi\)
\(462\) 0 0
\(463\) 18.6561 0.867024 0.433512 0.901148i \(-0.357274\pi\)
0.433512 + 0.901148i \(0.357274\pi\)
\(464\) 6.56035 0.304557
\(465\) 0 0
\(466\) 6.94824 0.321871
\(467\) 8.67074 0.401234 0.200617 0.979670i \(-0.435705\pi\)
0.200617 + 0.979670i \(0.435705\pi\)
\(468\) 0 0
\(469\) 52.7191 2.43434
\(470\) 0 0
\(471\) 0 0
\(472\) 4.65164 0.214109
\(473\) 15.9329 0.732594
\(474\) 0 0
\(475\) 0 0
\(476\) −29.5760 −1.35561
\(477\) 0 0
\(478\) 0.651639 0.0298053
\(479\) 20.4312 0.933523 0.466762 0.884383i \(-0.345421\pi\)
0.466762 + 0.884383i \(0.345421\pi\)
\(480\) 0 0
\(481\) −2.42160 −0.110416
\(482\) −2.80262 −0.127656
\(483\) 0 0
\(484\) 15.7164 0.714381
\(485\) 0 0
\(486\) 0 0
\(487\) 7.00611 0.317477 0.158739 0.987321i \(-0.449257\pi\)
0.158739 + 0.987321i \(0.449257\pi\)
\(488\) 10.1725 0.460486
\(489\) 0 0
\(490\) 0 0
\(491\) 15.5715 0.702733 0.351366 0.936238i \(-0.385717\pi\)
0.351366 + 0.936238i \(0.385717\pi\)
\(492\) 0 0
\(493\) −12.1268 −0.546164
\(494\) −0.263748 −0.0118666
\(495\) 0 0
\(496\) −7.87586 −0.353636
\(497\) −34.8286 −1.56228
\(498\) 0 0
\(499\) −1.61899 −0.0724757 −0.0362379 0.999343i \(-0.511537\pi\)
−0.0362379 + 0.999343i \(0.511537\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −1.27406 −0.0568642
\(503\) −39.3725 −1.75553 −0.877767 0.479088i \(-0.840968\pi\)
−0.877767 + 0.479088i \(0.840968\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −4.32582 −0.192306
\(507\) 0 0
\(508\) −1.46563 −0.0650267
\(509\) 8.63971 0.382948 0.191474 0.981498i \(-0.438673\pi\)
0.191474 + 0.981498i \(0.438673\pi\)
\(510\) 0 0
\(511\) 31.0732 1.37460
\(512\) 22.8302 1.00896
\(513\) 0 0
\(514\) −0.373283 −0.0164648
\(515\) 0 0
\(516\) 0 0
\(517\) −6.00955 −0.264300
\(518\) 15.1329 0.664903
\(519\) 0 0
\(520\) 0 0
\(521\) −30.7862 −1.34877 −0.674384 0.738381i \(-0.735591\pi\)
−0.674384 + 0.738381i \(0.735591\pi\)
\(522\) 0 0
\(523\) −17.4036 −0.761004 −0.380502 0.924780i \(-0.624249\pi\)
−0.380502 + 0.924780i \(0.624249\pi\)
\(524\) 14.4622 0.631784
\(525\) 0 0
\(526\) −7.33699 −0.319908
\(527\) 14.5585 0.634180
\(528\) 0 0
\(529\) 16.0518 0.697902
\(530\) 0 0
\(531\) 0 0
\(532\) −13.2311 −0.573641
\(533\) −0.249141 −0.0107915
\(534\) 0 0
\(535\) 0 0
\(536\) 28.3449 1.22431
\(537\) 0 0
\(538\) −1.43965 −0.0620676
\(539\) 5.79650 0.249673
\(540\) 0 0
\(541\) −9.87586 −0.424596 −0.212298 0.977205i \(-0.568095\pi\)
−0.212298 + 0.977205i \(0.568095\pi\)
\(542\) 5.78008 0.248275
\(543\) 0 0
\(544\) −24.3189 −1.04267
\(545\) 0 0
\(546\) 0 0
\(547\) 18.1465 0.775888 0.387944 0.921683i \(-0.373185\pi\)
0.387944 + 0.921683i \(0.373185\pi\)
\(548\) 21.7034 0.927123
\(549\) 0 0
\(550\) 0 0
\(551\) −5.42504 −0.231114
\(552\) 0 0
\(553\) 1.85352 0.0788196
\(554\) 4.03371 0.171376
\(555\) 0 0
\(556\) −25.4137 −1.07778
\(557\) 37.3725 1.58352 0.791762 0.610829i \(-0.209164\pi\)
0.791762 + 0.610829i \(0.209164\pi\)
\(558\) 0 0
\(559\) 2.69910 0.114160
\(560\) 0 0
\(561\) 0 0
\(562\) 6.86469 0.289569
\(563\) 22.8742 0.964034 0.482017 0.876162i \(-0.339904\pi\)
0.482017 + 0.876162i \(0.339904\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 9.86125 0.414499
\(567\) 0 0
\(568\) −18.7259 −0.785723
\(569\) −34.4147 −1.44274 −0.721370 0.692550i \(-0.756487\pi\)
−0.721370 + 0.692550i \(0.756487\pi\)
\(570\) 0 0
\(571\) 36.3449 1.52099 0.760494 0.649345i \(-0.224957\pi\)
0.760494 + 0.649345i \(0.224957\pi\)
\(572\) −0.651639 −0.0272464
\(573\) 0 0
\(574\) 1.55691 0.0649843
\(575\) 0 0
\(576\) 0 0
\(577\) −26.1725 −1.08957 −0.544787 0.838575i \(-0.683389\pi\)
−0.544787 + 0.838575i \(0.683389\pi\)
\(578\) 3.89572 0.162041
\(579\) 0 0
\(580\) 0 0
\(581\) −12.5896 −0.522303
\(582\) 0 0
\(583\) −2.11727 −0.0876882
\(584\) 16.7068 0.691334
\(585\) 0 0
\(586\) 14.7835 0.610701
\(587\) −17.8156 −0.735329 −0.367664 0.929959i \(-0.619842\pi\)
−0.367664 + 0.929959i \(0.619842\pi\)
\(588\) 0 0
\(589\) 6.51289 0.268359
\(590\) 0 0
\(591\) 0 0
\(592\) −26.4362 −1.08652
\(593\) −32.7018 −1.34290 −0.671451 0.741049i \(-0.734328\pi\)
−0.671451 + 0.741049i \(0.734328\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −39.3224 −1.61071
\(597\) 0 0
\(598\) −0.732814 −0.0299670
\(599\) −10.2784 −0.419962 −0.209981 0.977705i \(-0.567340\pi\)
−0.209981 + 0.977705i \(0.567340\pi\)
\(600\) 0 0
\(601\) 31.8398 1.29877 0.649386 0.760459i \(-0.275026\pi\)
0.649386 + 0.760459i \(0.275026\pi\)
\(602\) −16.8671 −0.687450
\(603\) 0 0
\(604\) −17.9931 −0.732130
\(605\) 0 0
\(606\) 0 0
\(607\) −10.2086 −0.414352 −0.207176 0.978304i \(-0.566427\pi\)
−0.207176 + 0.978304i \(0.566427\pi\)
\(608\) −10.8793 −0.441214
\(609\) 0 0
\(610\) 0 0
\(611\) −1.01805 −0.0411857
\(612\) 0 0
\(613\) 4.94137 0.199580 0.0997900 0.995009i \(-0.468183\pi\)
0.0997900 + 0.995009i \(0.468183\pi\)
\(614\) −4.02148 −0.162294
\(615\) 0 0
\(616\) 8.65164 0.348584
\(617\) 15.2526 0.614046 0.307023 0.951702i \(-0.400667\pi\)
0.307023 + 0.951702i \(0.400667\pi\)
\(618\) 0 0
\(619\) −20.6543 −0.830167 −0.415084 0.909783i \(-0.636248\pi\)
−0.415084 + 0.909783i \(0.636248\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 2.58633 0.103702
\(623\) 13.6190 0.545633
\(624\) 0 0
\(625\) 0 0
\(626\) −2.57946 −0.103096
\(627\) 0 0
\(628\) 33.3675 1.33151
\(629\) 48.8674 1.94847
\(630\) 0 0
\(631\) 5.83709 0.232371 0.116185 0.993228i \(-0.462933\pi\)
0.116185 + 0.993228i \(0.462933\pi\)
\(632\) 0.996562 0.0396411
\(633\) 0 0
\(634\) −5.32688 −0.211557
\(635\) 0 0
\(636\) 0 0
\(637\) 0.981954 0.0389064
\(638\) 1.66968 0.0661034
\(639\) 0 0
\(640\) 0 0
\(641\) −42.6328 −1.68390 −0.841948 0.539559i \(-0.818591\pi\)
−0.841948 + 0.539559i \(0.818591\pi\)
\(642\) 0 0
\(643\) −30.7259 −1.21171 −0.605856 0.795574i \(-0.707169\pi\)
−0.605856 + 0.795574i \(0.707169\pi\)
\(644\) −36.7620 −1.44863
\(645\) 0 0
\(646\) 5.32238 0.209406
\(647\) 18.3595 0.721788 0.360894 0.932607i \(-0.382472\pi\)
0.360894 + 0.932607i \(0.382472\pi\)
\(648\) 0 0
\(649\) −3.84664 −0.150994
\(650\) 0 0
\(651\) 0 0
\(652\) 15.1950 0.595082
\(653\) −35.5811 −1.39240 −0.696198 0.717850i \(-0.745126\pi\)
−0.696198 + 0.717850i \(0.745126\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −2.71982 −0.106191
\(657\) 0 0
\(658\) 6.36191 0.248013
\(659\) 6.38101 0.248569 0.124285 0.992247i \(-0.460336\pi\)
0.124285 + 0.992247i \(0.460336\pi\)
\(660\) 0 0
\(661\) 30.9053 1.20208 0.601038 0.799220i \(-0.294754\pi\)
0.601038 + 0.799220i \(0.294754\pi\)
\(662\) 15.9379 0.619445
\(663\) 0 0
\(664\) −6.76891 −0.262684
\(665\) 0 0
\(666\) 0 0
\(667\) −15.0732 −0.583638
\(668\) −8.99656 −0.348087
\(669\) 0 0
\(670\) 0 0
\(671\) −8.41205 −0.324744
\(672\) 0 0
\(673\) −36.7259 −1.41568 −0.707840 0.706372i \(-0.750330\pi\)
−0.707840 + 0.706372i \(0.750330\pi\)
\(674\) 3.19051 0.122894
\(675\) 0 0
\(676\) 23.0096 0.884983
\(677\) −5.66281 −0.217639 −0.108820 0.994062i \(-0.534707\pi\)
−0.108820 + 0.994062i \(0.534707\pi\)
\(678\) 0 0
\(679\) −23.7846 −0.912768
\(680\) 0 0
\(681\) 0 0
\(682\) −2.00450 −0.0767561
\(683\) −29.1690 −1.11612 −0.558061 0.829800i \(-0.688454\pi\)
−0.558061 + 0.829800i \(0.688454\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 4.76203 0.181815
\(687\) 0 0
\(688\) 29.4656 1.12337
\(689\) −0.358675 −0.0136644
\(690\) 0 0
\(691\) 1.68879 0.0642445 0.0321223 0.999484i \(-0.489773\pi\)
0.0321223 + 0.999484i \(0.489773\pi\)
\(692\) −33.5500 −1.27538
\(693\) 0 0
\(694\) 9.03715 0.343046
\(695\) 0 0
\(696\) 0 0
\(697\) 5.02760 0.190434
\(698\) −3.08012 −0.116584
\(699\) 0 0
\(700\) 0 0
\(701\) 20.3810 0.769780 0.384890 0.922962i \(-0.374239\pi\)
0.384890 + 0.922962i \(0.374239\pi\)
\(702\) 0 0
\(703\) 21.8613 0.824513
\(704\) −4.65164 −0.175315
\(705\) 0 0
\(706\) −7.52770 −0.283309
\(707\) −36.8647 −1.38644
\(708\) 0 0
\(709\) −40.8363 −1.53364 −0.766820 0.641862i \(-0.778162\pi\)
−0.766820 + 0.641862i \(0.778162\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 7.32238 0.274418
\(713\) 18.0958 0.677692
\(714\) 0 0
\(715\) 0 0
\(716\) −5.07774 −0.189764
\(717\) 0 0
\(718\) 5.71027 0.213105
\(719\) −9.14486 −0.341046 −0.170523 0.985354i \(-0.554546\pi\)
−0.170523 + 0.985354i \(0.554546\pi\)
\(720\) 0 0
\(721\) 43.9164 1.63553
\(722\) −6.56197 −0.244211
\(723\) 0 0
\(724\) −23.9018 −0.888305
\(725\) 0 0
\(726\) 0 0
\(727\) 2.13875 0.0793218 0.0396609 0.999213i \(-0.487372\pi\)
0.0396609 + 0.999213i \(0.487372\pi\)
\(728\) 1.46563 0.0543198
\(729\) 0 0
\(730\) 0 0
\(731\) −54.4672 −2.01454
\(732\) 0 0
\(733\) −15.3388 −0.566552 −0.283276 0.959038i \(-0.591421\pi\)
−0.283276 + 0.959038i \(0.591421\pi\)
\(734\) −3.01117 −0.111144
\(735\) 0 0
\(736\) −30.2277 −1.11421
\(737\) −23.4396 −0.863411
\(738\) 0 0
\(739\) −15.8111 −0.581621 −0.290811 0.956781i \(-0.593925\pi\)
−0.290811 + 0.956781i \(0.593925\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 2.24141 0.0822847
\(743\) −20.2017 −0.741128 −0.370564 0.928807i \(-0.620836\pi\)
−0.370564 + 0.928807i \(0.620836\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 3.54918 0.129945
\(747\) 0 0
\(748\) 13.1499 0.480809
\(749\) −30.8363 −1.12673
\(750\) 0 0
\(751\) 32.9053 1.20073 0.600365 0.799726i \(-0.295022\pi\)
0.600365 + 0.799726i \(0.295022\pi\)
\(752\) −11.1138 −0.405280
\(753\) 0 0
\(754\) 0.282852 0.0103009
\(755\) 0 0
\(756\) 0 0
\(757\) −11.3009 −0.410738 −0.205369 0.978685i \(-0.565839\pi\)
−0.205369 + 0.978685i \(0.565839\pi\)
\(758\) −0.0292158 −0.00106117
\(759\) 0 0
\(760\) 0 0
\(761\) 2.93449 0.106375 0.0531876 0.998585i \(-0.483062\pi\)
0.0531876 + 0.998585i \(0.483062\pi\)
\(762\) 0 0
\(763\) 0.824101 0.0298344
\(764\) 24.0000 0.868290
\(765\) 0 0
\(766\) 11.2648 0.407014
\(767\) −0.651639 −0.0235293
\(768\) 0 0
\(769\) −22.7785 −0.821412 −0.410706 0.911768i \(-0.634718\pi\)
−0.410706 + 0.911768i \(0.634718\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −14.2536 −0.513000
\(773\) 10.5553 0.379648 0.189824 0.981818i \(-0.439208\pi\)
0.189824 + 0.981818i \(0.439208\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −12.7880 −0.459063
\(777\) 0 0
\(778\) 11.2311 0.402654
\(779\) 2.24914 0.0805838
\(780\) 0 0
\(781\) 15.4853 0.554107
\(782\) 14.7880 0.528818
\(783\) 0 0
\(784\) 10.7198 0.382851
\(785\) 0 0
\(786\) 0 0
\(787\) 23.0682 0.822292 0.411146 0.911570i \(-0.365129\pi\)
0.411146 + 0.911570i \(0.365129\pi\)
\(788\) −24.8602 −0.885608
\(789\) 0 0
\(790\) 0 0
\(791\) −30.5827 −1.08740
\(792\) 0 0
\(793\) −1.42504 −0.0506047
\(794\) −6.88617 −0.244381
\(795\) 0 0
\(796\) 8.65164 0.306649
\(797\) 0.0360915 0.00127843 0.000639213 1.00000i \(-0.499797\pi\)
0.000639213 1.00000i \(0.499797\pi\)
\(798\) 0 0
\(799\) 20.5439 0.726792
\(800\) 0 0
\(801\) 0 0
\(802\) 5.32926 0.188183
\(803\) −13.8156 −0.487542
\(804\) 0 0
\(805\) 0 0
\(806\) −0.339571 −0.0119609
\(807\) 0 0
\(808\) −19.8207 −0.697288
\(809\) −33.1430 −1.16525 −0.582624 0.812742i \(-0.697974\pi\)
−0.582624 + 0.812742i \(0.697974\pi\)
\(810\) 0 0
\(811\) −1.10428 −0.0387764 −0.0193882 0.999812i \(-0.506172\pi\)
−0.0193882 + 0.999812i \(0.506172\pi\)
\(812\) 14.1894 0.497952
\(813\) 0 0
\(814\) −6.72832 −0.235827
\(815\) 0 0
\(816\) 0 0
\(817\) −24.3664 −0.852473
\(818\) −15.9525 −0.557767
\(819\) 0 0
\(820\) 0 0
\(821\) 7.42504 0.259136 0.129568 0.991571i \(-0.458641\pi\)
0.129568 + 0.991571i \(0.458641\pi\)
\(822\) 0 0
\(823\) −42.1656 −1.46980 −0.734900 0.678176i \(-0.762771\pi\)
−0.734900 + 0.678176i \(0.762771\pi\)
\(824\) 23.6121 0.822567
\(825\) 0 0
\(826\) 4.07218 0.141689
\(827\) −54.4863 −1.89468 −0.947338 0.320235i \(-0.896238\pi\)
−0.947338 + 0.320235i \(0.896238\pi\)
\(828\) 0 0
\(829\) 24.0388 0.834901 0.417450 0.908700i \(-0.362924\pi\)
0.417450 + 0.908700i \(0.362924\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −0.788009 −0.0273193
\(833\) −19.8156 −0.686570
\(834\) 0 0
\(835\) 0 0
\(836\) 5.88273 0.203459
\(837\) 0 0
\(838\) 0.934491 0.0322815
\(839\) 13.6190 0.470180 0.235090 0.971974i \(-0.424462\pi\)
0.235090 + 0.971974i \(0.424462\pi\)
\(840\) 0 0
\(841\) −23.1820 −0.799380
\(842\) −2.67074 −0.0920399
\(843\) 0 0
\(844\) 18.8793 0.649852
\(845\) 0 0
\(846\) 0 0
\(847\) 29.2311 1.00439
\(848\) −3.91559 −0.134462
\(849\) 0 0
\(850\) 0 0
\(851\) 60.7405 2.08216
\(852\) 0 0
\(853\) 6.93449 0.237432 0.118716 0.992928i \(-0.462122\pi\)
0.118716 + 0.992928i \(0.462122\pi\)
\(854\) 8.90528 0.304732
\(855\) 0 0
\(856\) −16.5795 −0.566674
\(857\) −34.9345 −1.19334 −0.596670 0.802487i \(-0.703510\pi\)
−0.596670 + 0.802487i \(0.703510\pi\)
\(858\) 0 0
\(859\) 4.39057 0.149804 0.0749021 0.997191i \(-0.476136\pi\)
0.0749021 + 0.997191i \(0.476136\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −0.824101 −0.0280690
\(863\) 15.4611 0.526303 0.263152 0.964755i \(-0.415238\pi\)
0.263152 + 0.964755i \(0.415238\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 16.0337 0.544848
\(867\) 0 0
\(868\) −17.0348 −0.578198
\(869\) −0.824101 −0.0279557
\(870\) 0 0
\(871\) −3.97078 −0.134545
\(872\) 0.443086 0.0150048
\(873\) 0 0
\(874\) 6.61555 0.223774
\(875\) 0 0
\(876\) 0 0
\(877\) 55.2338 1.86511 0.932556 0.361025i \(-0.117573\pi\)
0.932556 + 0.361025i \(0.117573\pi\)
\(878\) −2.03609 −0.0687148
\(879\) 0 0
\(880\) 0 0
\(881\) 28.4431 0.958272 0.479136 0.877741i \(-0.340950\pi\)
0.479136 + 0.877741i \(0.340950\pi\)
\(882\) 0 0
\(883\) 24.1319 0.812102 0.406051 0.913850i \(-0.366905\pi\)
0.406051 + 0.913850i \(0.366905\pi\)
\(884\) 2.22766 0.0749242
\(885\) 0 0
\(886\) 17.4137 0.585024
\(887\) −2.06025 −0.0691765 −0.0345882 0.999402i \(-0.511012\pi\)
−0.0345882 + 0.999402i \(0.511012\pi\)
\(888\) 0 0
\(889\) −2.72594 −0.0914250
\(890\) 0 0
\(891\) 0 0
\(892\) −32.0000 −1.07144
\(893\) 9.19051 0.307549
\(894\) 0 0
\(895\) 0 0
\(896\) 36.9244 1.23356
\(897\) 0 0
\(898\) 7.50496 0.250444
\(899\) −6.98463 −0.232950
\(900\) 0 0
\(901\) 7.23797 0.241132
\(902\) −0.692226 −0.0230486
\(903\) 0 0
\(904\) −16.4431 −0.546889
\(905\) 0 0
\(906\) 0 0
\(907\) 19.5208 0.648178 0.324089 0.946027i \(-0.394942\pi\)
0.324089 + 0.946027i \(0.394942\pi\)
\(908\) 1.61899 0.0537279
\(909\) 0 0
\(910\) 0 0
\(911\) 39.9785 1.32455 0.662274 0.749262i \(-0.269592\pi\)
0.662274 + 0.749262i \(0.269592\pi\)
\(912\) 0 0
\(913\) 5.59750 0.185250
\(914\) 13.8568 0.458341
\(915\) 0 0
\(916\) 13.2051 0.436309
\(917\) 26.8984 0.888263
\(918\) 0 0
\(919\) −26.4508 −0.872532 −0.436266 0.899818i \(-0.643699\pi\)
−0.436266 + 0.899818i \(0.643699\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −10.2966 −0.339101
\(923\) 2.62328 0.0863463
\(924\) 0 0
\(925\) 0 0
\(926\) 8.78113 0.288566
\(927\) 0 0
\(928\) 11.6673 0.382998
\(929\) −33.2913 −1.09225 −0.546127 0.837703i \(-0.683898\pi\)
−0.546127 + 0.837703i \(0.683898\pi\)
\(930\) 0 0
\(931\) −8.86469 −0.290528
\(932\) −26.2536 −0.859966
\(933\) 0 0
\(934\) 4.08117 0.133540
\(935\) 0 0
\(936\) 0 0
\(937\) 38.7811 1.26692 0.633462 0.773774i \(-0.281633\pi\)
0.633462 + 0.773774i \(0.281633\pi\)
\(938\) 24.8140 0.810205
\(939\) 0 0
\(940\) 0 0
\(941\) −15.6482 −0.510117 −0.255058 0.966926i \(-0.582095\pi\)
−0.255058 + 0.966926i \(0.582095\pi\)
\(942\) 0 0
\(943\) 6.24914 0.203500
\(944\) −7.11383 −0.231535
\(945\) 0 0
\(946\) 7.49934 0.243825
\(947\) −47.5500 −1.54517 −0.772584 0.634912i \(-0.781036\pi\)
−0.772584 + 0.634912i \(0.781036\pi\)
\(948\) 0 0
\(949\) −2.34043 −0.0759735
\(950\) 0 0
\(951\) 0 0
\(952\) −29.5760 −0.958564
\(953\) −29.6819 −0.961491 −0.480746 0.876860i \(-0.659634\pi\)
−0.480746 + 0.876860i \(0.659634\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −2.46219 −0.0796329
\(957\) 0 0
\(958\) 9.61661 0.310698
\(959\) 40.3664 1.30350
\(960\) 0 0
\(961\) −22.6148 −0.729509
\(962\) −1.13981 −0.0367489
\(963\) 0 0
\(964\) 10.5896 0.341067
\(965\) 0 0
\(966\) 0 0
\(967\) 1.21199 0.0389750 0.0194875 0.999810i \(-0.493797\pi\)
0.0194875 + 0.999810i \(0.493797\pi\)
\(968\) 15.7164 0.505144
\(969\) 0 0
\(970\) 0 0
\(971\) −12.5362 −0.402306 −0.201153 0.979560i \(-0.564469\pi\)
−0.201153 + 0.979560i \(0.564469\pi\)
\(972\) 0 0
\(973\) −47.2672 −1.51532
\(974\) 3.29766 0.105664
\(975\) 0 0
\(976\) −15.5569 −0.497965
\(977\) 15.2069 0.486513 0.243256 0.969962i \(-0.421784\pi\)
0.243256 + 0.969962i \(0.421784\pi\)
\(978\) 0 0
\(979\) −6.05520 −0.193525
\(980\) 0 0
\(981\) 0 0
\(982\) 7.32926 0.233886
\(983\) 39.0303 1.24487 0.622436 0.782671i \(-0.286143\pi\)
0.622436 + 0.782671i \(0.286143\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −5.70789 −0.181776
\(987\) 0 0
\(988\) 0.996562 0.0317049
\(989\) −67.7010 −2.15277
\(990\) 0 0
\(991\) −8.91539 −0.283207 −0.141603 0.989923i \(-0.545226\pi\)
−0.141603 + 0.989923i \(0.545226\pi\)
\(992\) −14.0069 −0.444719
\(993\) 0 0
\(994\) −16.3932 −0.519962
\(995\) 0 0
\(996\) 0 0
\(997\) 33.6336 1.06519 0.532593 0.846371i \(-0.321218\pi\)
0.532593 + 0.846371i \(0.321218\pi\)
\(998\) −0.762030 −0.0241216
\(999\) 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 9225.2.a.bx.1.2 3
3.2 odd 2 3075.2.a.t.1.2 3
5.4 even 2 369.2.a.e.1.2 3
15.14 odd 2 123.2.a.d.1.2 3
20.19 odd 2 5904.2.a.bd.1.1 3
60.59 even 2 1968.2.a.w.1.3 3
105.104 even 2 6027.2.a.s.1.2 3
120.29 odd 2 7872.2.a.bx.1.1 3
120.59 even 2 7872.2.a.bs.1.1 3
615.614 odd 2 5043.2.a.n.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
123.2.a.d.1.2 3 15.14 odd 2
369.2.a.e.1.2 3 5.4 even 2
1968.2.a.w.1.3 3 60.59 even 2
3075.2.a.t.1.2 3 3.2 odd 2
5043.2.a.n.1.2 3 615.614 odd 2
5904.2.a.bd.1.1 3 20.19 odd 2
6027.2.a.s.1.2 3 105.104 even 2
7872.2.a.bs.1.1 3 120.59 even 2
7872.2.a.bx.1.1 3 120.29 odd 2
9225.2.a.bx.1.2 3 1.1 even 1 trivial