Properties

Label 1350.4.c.f.649.1
Level $1350$
Weight $4$
Character 1350.649
Analytic conductor $79.653$
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] = [1350,4,Mod(649,1350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1350, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1350.649");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1350.c (of order \(2\), degree \(1\), not 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: \(79.6525785077\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 270)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1350.649
Dual form 1350.4.c.f.649.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} -4.00000 q^{4} +13.0000i q^{7} +8.00000i q^{8} +O(q^{10})\) \(q-2.00000i q^{2} -4.00000 q^{4} +13.0000i q^{7} +8.00000i q^{8} -30.0000 q^{11} -61.0000i q^{13} +26.0000 q^{14} +16.0000 q^{16} -12.0000i q^{17} +49.0000 q^{19} +60.0000i q^{22} +18.0000i q^{23} -122.000 q^{26} -52.0000i q^{28} +186.000 q^{29} -160.000 q^{31} -32.0000i q^{32} -24.0000 q^{34} +91.0000i q^{37} -98.0000i q^{38} +378.000 q^{41} -268.000i q^{43} +120.000 q^{44} +36.0000 q^{46} -144.000i q^{47} +174.000 q^{49} +244.000i q^{52} +570.000i q^{53} -104.000 q^{56} -372.000i q^{58} -204.000 q^{59} -877.000 q^{61} +320.000i q^{62} -64.0000 q^{64} +187.000i q^{67} +48.0000i q^{68} -606.000 q^{71} +431.000i q^{73} +182.000 q^{74} -196.000 q^{76} -390.000i q^{77} -1151.00 q^{79} -756.000i q^{82} +102.000i q^{83} -536.000 q^{86} -240.000i q^{88} -984.000 q^{89} +793.000 q^{91} -72.0000i q^{92} -288.000 q^{94} +265.000i q^{97} -348.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} - 60 q^{11} + 52 q^{14} + 32 q^{16} + 98 q^{19} - 244 q^{26} + 372 q^{29} - 320 q^{31} - 48 q^{34} + 756 q^{41} + 240 q^{44} + 72 q^{46} + 348 q^{49} - 208 q^{56} - 408 q^{59} - 1754 q^{61} - 128 q^{64} - 1212 q^{71} + 364 q^{74} - 392 q^{76} - 2302 q^{79} - 1072 q^{86} - 1968 q^{89} + 1586 q^{91} - 576 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 0.707107i
\(3\) 0 0
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 13.0000i 0.701934i 0.936388 + 0.350967i \(0.114147\pi\)
−0.936388 + 0.350967i \(0.885853\pi\)
\(8\) 8.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −30.0000 −0.822304 −0.411152 0.911567i \(-0.634873\pi\)
−0.411152 + 0.911567i \(0.634873\pi\)
\(12\) 0 0
\(13\) − 61.0000i − 1.30141i −0.759330 0.650706i \(-0.774473\pi\)
0.759330 0.650706i \(-0.225527\pi\)
\(14\) 26.0000 0.496342
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) − 12.0000i − 0.171202i −0.996330 0.0856008i \(-0.972719\pi\)
0.996330 0.0856008i \(-0.0272810\pi\)
\(18\) 0 0
\(19\) 49.0000 0.591651 0.295826 0.955242i \(-0.404405\pi\)
0.295826 + 0.955242i \(0.404405\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 60.0000i 0.581456i
\(23\) 18.0000i 0.163185i 0.996666 + 0.0815926i \(0.0260006\pi\)
−0.996666 + 0.0815926i \(0.973999\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −122.000 −0.920237
\(27\) 0 0
\(28\) − 52.0000i − 0.350967i
\(29\) 186.000 1.19101 0.595506 0.803351i \(-0.296952\pi\)
0.595506 + 0.803351i \(0.296952\pi\)
\(30\) 0 0
\(31\) −160.000 −0.926995 −0.463498 0.886098i \(-0.653406\pi\)
−0.463498 + 0.886098i \(0.653406\pi\)
\(32\) − 32.0000i − 0.176777i
\(33\) 0 0
\(34\) −24.0000 −0.121058
\(35\) 0 0
\(36\) 0 0
\(37\) 91.0000i 0.404333i 0.979351 + 0.202166i \(0.0647982\pi\)
−0.979351 + 0.202166i \(0.935202\pi\)
\(38\) − 98.0000i − 0.418361i
\(39\) 0 0
\(40\) 0 0
\(41\) 378.000 1.43985 0.719923 0.694054i \(-0.244177\pi\)
0.719923 + 0.694054i \(0.244177\pi\)
\(42\) 0 0
\(43\) − 268.000i − 0.950456i −0.879863 0.475228i \(-0.842366\pi\)
0.879863 0.475228i \(-0.157634\pi\)
\(44\) 120.000 0.411152
\(45\) 0 0
\(46\) 36.0000 0.115389
\(47\) − 144.000i − 0.446906i −0.974715 0.223453i \(-0.928267\pi\)
0.974715 0.223453i \(-0.0717328\pi\)
\(48\) 0 0
\(49\) 174.000 0.507289
\(50\) 0 0
\(51\) 0 0
\(52\) 244.000i 0.650706i
\(53\) 570.000i 1.47727i 0.674103 + 0.738637i \(0.264530\pi\)
−0.674103 + 0.738637i \(0.735470\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −104.000 −0.248171
\(57\) 0 0
\(58\) − 372.000i − 0.842172i
\(59\) −204.000 −0.450145 −0.225072 0.974342i \(-0.572262\pi\)
−0.225072 + 0.974342i \(0.572262\pi\)
\(60\) 0 0
\(61\) −877.000 −1.84079 −0.920396 0.390987i \(-0.872134\pi\)
−0.920396 + 0.390987i \(0.872134\pi\)
\(62\) 320.000i 0.655485i
\(63\) 0 0
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 187.000i 0.340980i 0.985359 + 0.170490i \(0.0545351\pi\)
−0.985359 + 0.170490i \(0.945465\pi\)
\(68\) 48.0000i 0.0856008i
\(69\) 0 0
\(70\) 0 0
\(71\) −606.000 −1.01294 −0.506472 0.862257i \(-0.669050\pi\)
−0.506472 + 0.862257i \(0.669050\pi\)
\(72\) 0 0
\(73\) 431.000i 0.691024i 0.938414 + 0.345512i \(0.112295\pi\)
−0.938414 + 0.345512i \(0.887705\pi\)
\(74\) 182.000 0.285906
\(75\) 0 0
\(76\) −196.000 −0.295826
\(77\) − 390.000i − 0.577203i
\(78\) 0 0
\(79\) −1151.00 −1.63921 −0.819605 0.572929i \(-0.805807\pi\)
−0.819605 + 0.572929i \(0.805807\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 756.000i − 1.01812i
\(83\) 102.000i 0.134891i 0.997723 + 0.0674455i \(0.0214849\pi\)
−0.997723 + 0.0674455i \(0.978515\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −536.000 −0.672074
\(87\) 0 0
\(88\) − 240.000i − 0.290728i
\(89\) −984.000 −1.17195 −0.585976 0.810328i \(-0.699289\pi\)
−0.585976 + 0.810328i \(0.699289\pi\)
\(90\) 0 0
\(91\) 793.000 0.913505
\(92\) − 72.0000i − 0.0815926i
\(93\) 0 0
\(94\) −288.000 −0.316010
\(95\) 0 0
\(96\) 0 0
\(97\) 265.000i 0.277388i 0.990335 + 0.138694i \(0.0442905\pi\)
−0.990335 + 0.138694i \(0.955709\pi\)
\(98\) − 348.000i − 0.358707i
\(99\) 0 0
\(100\) 0 0
\(101\) −1248.00 −1.22951 −0.614756 0.788718i \(-0.710745\pi\)
−0.614756 + 0.788718i \(0.710745\pi\)
\(102\) 0 0
\(103\) − 1225.00i − 1.17187i −0.810357 0.585936i \(-0.800727\pi\)
0.810357 0.585936i \(-0.199273\pi\)
\(104\) 488.000 0.460119
\(105\) 0 0
\(106\) 1140.00 1.04459
\(107\) 78.0000i 0.0704724i 0.999379 + 0.0352362i \(0.0112183\pi\)
−0.999379 + 0.0352362i \(0.988782\pi\)
\(108\) 0 0
\(109\) −2198.00 −1.93147 −0.965735 0.259530i \(-0.916432\pi\)
−0.965735 + 0.259530i \(0.916432\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 208.000i 0.175484i
\(113\) 1986.00i 1.65334i 0.562689 + 0.826669i \(0.309767\pi\)
−0.562689 + 0.826669i \(0.690233\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −744.000 −0.595506
\(117\) 0 0
\(118\) 408.000i 0.318300i
\(119\) 156.000 0.120172
\(120\) 0 0
\(121\) −431.000 −0.323817
\(122\) 1754.00i 1.30164i
\(123\) 0 0
\(124\) 640.000 0.463498
\(125\) 0 0
\(126\) 0 0
\(127\) − 2792.00i − 1.95079i −0.220471 0.975393i \(-0.570760\pi\)
0.220471 0.975393i \(-0.429240\pi\)
\(128\) 128.000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −708.000 −0.472200 −0.236100 0.971729i \(-0.575869\pi\)
−0.236100 + 0.971729i \(0.575869\pi\)
\(132\) 0 0
\(133\) 637.000i 0.415300i
\(134\) 374.000 0.241110
\(135\) 0 0
\(136\) 96.0000 0.0605289
\(137\) 1686.00i 1.05142i 0.850663 + 0.525711i \(0.176201\pi\)
−0.850663 + 0.525711i \(0.823799\pi\)
\(138\) 0 0
\(139\) 307.000 0.187334 0.0936669 0.995604i \(-0.470141\pi\)
0.0936669 + 0.995604i \(0.470141\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1212.00i 0.716259i
\(143\) 1830.00i 1.07016i
\(144\) 0 0
\(145\) 0 0
\(146\) 862.000 0.488628
\(147\) 0 0
\(148\) − 364.000i − 0.202166i
\(149\) 1812.00 0.996274 0.498137 0.867098i \(-0.334018\pi\)
0.498137 + 0.867098i \(0.334018\pi\)
\(150\) 0 0
\(151\) 203.000 0.109403 0.0547017 0.998503i \(-0.482579\pi\)
0.0547017 + 0.998503i \(0.482579\pi\)
\(152\) 392.000i 0.209180i
\(153\) 0 0
\(154\) −780.000 −0.408144
\(155\) 0 0
\(156\) 0 0
\(157\) 214.000i 0.108784i 0.998520 + 0.0543919i \(0.0173220\pi\)
−0.998520 + 0.0543919i \(0.982678\pi\)
\(158\) 2302.00i 1.15910i
\(159\) 0 0
\(160\) 0 0
\(161\) −234.000 −0.114545
\(162\) 0 0
\(163\) − 673.000i − 0.323395i −0.986840 0.161698i \(-0.948303\pi\)
0.986840 0.161698i \(-0.0516969\pi\)
\(164\) −1512.00 −0.719923
\(165\) 0 0
\(166\) 204.000 0.0953824
\(167\) 3696.00i 1.71261i 0.516474 + 0.856303i \(0.327244\pi\)
−0.516474 + 0.856303i \(0.672756\pi\)
\(168\) 0 0
\(169\) −1524.00 −0.693673
\(170\) 0 0
\(171\) 0 0
\(172\) 1072.00i 0.475228i
\(173\) 3132.00i 1.37643i 0.725509 + 0.688213i \(0.241604\pi\)
−0.725509 + 0.688213i \(0.758396\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −480.000 −0.205576
\(177\) 0 0
\(178\) 1968.00i 0.828696i
\(179\) −510.000 −0.212956 −0.106478 0.994315i \(-0.533957\pi\)
−0.106478 + 0.994315i \(0.533957\pi\)
\(180\) 0 0
\(181\) −1087.00 −0.446387 −0.223194 0.974774i \(-0.571648\pi\)
−0.223194 + 0.974774i \(0.571648\pi\)
\(182\) − 1586.00i − 0.645946i
\(183\) 0 0
\(184\) −144.000 −0.0576947
\(185\) 0 0
\(186\) 0 0
\(187\) 360.000i 0.140780i
\(188\) 576.000i 0.223453i
\(189\) 0 0
\(190\) 0 0
\(191\) −4056.00 −1.53655 −0.768277 0.640117i \(-0.778886\pi\)
−0.768277 + 0.640117i \(0.778886\pi\)
\(192\) 0 0
\(193\) 473.000i 0.176411i 0.996102 + 0.0882054i \(0.0281132\pi\)
−0.996102 + 0.0882054i \(0.971887\pi\)
\(194\) 530.000 0.196143
\(195\) 0 0
\(196\) −696.000 −0.253644
\(197\) − 2556.00i − 0.924403i −0.886775 0.462202i \(-0.847060\pi\)
0.886775 0.462202i \(-0.152940\pi\)
\(198\) 0 0
\(199\) 2923.00 1.04124 0.520618 0.853790i \(-0.325702\pi\)
0.520618 + 0.853790i \(0.325702\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 2496.00i 0.869396i
\(203\) 2418.00i 0.836011i
\(204\) 0 0
\(205\) 0 0
\(206\) −2450.00 −0.828639
\(207\) 0 0
\(208\) − 976.000i − 0.325353i
\(209\) −1470.00 −0.486517
\(210\) 0 0
\(211\) −3175.00 −1.03591 −0.517953 0.855409i \(-0.673306\pi\)
−0.517953 + 0.855409i \(0.673306\pi\)
\(212\) − 2280.00i − 0.738637i
\(213\) 0 0
\(214\) 156.000 0.0498315
\(215\) 0 0
\(216\) 0 0
\(217\) − 2080.00i − 0.650689i
\(218\) 4396.00i 1.36576i
\(219\) 0 0
\(220\) 0 0
\(221\) −732.000 −0.222804
\(222\) 0 0
\(223\) − 2176.00i − 0.653434i −0.945122 0.326717i \(-0.894058\pi\)
0.945122 0.326717i \(-0.105942\pi\)
\(224\) 416.000 0.124086
\(225\) 0 0
\(226\) 3972.00 1.16909
\(227\) 3834.00i 1.12102i 0.828148 + 0.560510i \(0.189395\pi\)
−0.828148 + 0.560510i \(0.810605\pi\)
\(228\) 0 0
\(229\) 3202.00 0.923992 0.461996 0.886882i \(-0.347133\pi\)
0.461996 + 0.886882i \(0.347133\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1488.00i 0.421086i
\(233\) 4152.00i 1.16741i 0.811966 + 0.583705i \(0.198398\pi\)
−0.811966 + 0.583705i \(0.801602\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 816.000 0.225072
\(237\) 0 0
\(238\) − 312.000i − 0.0849746i
\(239\) 5466.00 1.47936 0.739678 0.672961i \(-0.234978\pi\)
0.739678 + 0.672961i \(0.234978\pi\)
\(240\) 0 0
\(241\) −943.000 −0.252050 −0.126025 0.992027i \(-0.540222\pi\)
−0.126025 + 0.992027i \(0.540222\pi\)
\(242\) 862.000i 0.228973i
\(243\) 0 0
\(244\) 3508.00 0.920396
\(245\) 0 0
\(246\) 0 0
\(247\) − 2989.00i − 0.769982i
\(248\) − 1280.00i − 0.327742i
\(249\) 0 0
\(250\) 0 0
\(251\) −7290.00 −1.83323 −0.916615 0.399771i \(-0.869090\pi\)
−0.916615 + 0.399771i \(0.869090\pi\)
\(252\) 0 0
\(253\) − 540.000i − 0.134188i
\(254\) −5584.00 −1.37941
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) − 312.000i − 0.0757277i −0.999283 0.0378639i \(-0.987945\pi\)
0.999283 0.0378639i \(-0.0120553\pi\)
\(258\) 0 0
\(259\) −1183.00 −0.283815
\(260\) 0 0
\(261\) 0 0
\(262\) 1416.00i 0.333896i
\(263\) − 8004.00i − 1.87661i −0.345812 0.938304i \(-0.612397\pi\)
0.345812 0.938304i \(-0.387603\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 1274.00 0.293661
\(267\) 0 0
\(268\) − 748.000i − 0.170490i
\(269\) 324.000 0.0734373 0.0367186 0.999326i \(-0.488309\pi\)
0.0367186 + 0.999326i \(0.488309\pi\)
\(270\) 0 0
\(271\) −7849.00 −1.75938 −0.879692 0.475545i \(-0.842251\pi\)
−0.879692 + 0.475545i \(0.842251\pi\)
\(272\) − 192.000i − 0.0428004i
\(273\) 0 0
\(274\) 3372.00 0.743467
\(275\) 0 0
\(276\) 0 0
\(277\) 5758.00i 1.24897i 0.781037 + 0.624485i \(0.214691\pi\)
−0.781037 + 0.624485i \(0.785309\pi\)
\(278\) − 614.000i − 0.132465i
\(279\) 0 0
\(280\) 0 0
\(281\) −2688.00 −0.570650 −0.285325 0.958431i \(-0.592102\pi\)
−0.285325 + 0.958431i \(0.592102\pi\)
\(282\) 0 0
\(283\) 3260.00i 0.684759i 0.939562 + 0.342380i \(0.111233\pi\)
−0.939562 + 0.342380i \(0.888767\pi\)
\(284\) 2424.00 0.506472
\(285\) 0 0
\(286\) 3660.00 0.756714
\(287\) 4914.00i 1.01068i
\(288\) 0 0
\(289\) 4769.00 0.970690
\(290\) 0 0
\(291\) 0 0
\(292\) − 1724.00i − 0.345512i
\(293\) 5922.00i 1.18077i 0.807120 + 0.590387i \(0.201025\pi\)
−0.807120 + 0.590387i \(0.798975\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −728.000 −0.142953
\(297\) 0 0
\(298\) − 3624.00i − 0.704472i
\(299\) 1098.00 0.212371
\(300\) 0 0
\(301\) 3484.00 0.667158
\(302\) − 406.000i − 0.0773599i
\(303\) 0 0
\(304\) 784.000 0.147913
\(305\) 0 0
\(306\) 0 0
\(307\) − 3728.00i − 0.693056i −0.938040 0.346528i \(-0.887361\pi\)
0.938040 0.346528i \(-0.112639\pi\)
\(308\) 1560.00i 0.288601i
\(309\) 0 0
\(310\) 0 0
\(311\) −732.000 −0.133466 −0.0667330 0.997771i \(-0.521258\pi\)
−0.0667330 + 0.997771i \(0.521258\pi\)
\(312\) 0 0
\(313\) 5357.00i 0.967398i 0.875234 + 0.483699i \(0.160707\pi\)
−0.875234 + 0.483699i \(0.839293\pi\)
\(314\) 428.000 0.0769218
\(315\) 0 0
\(316\) 4604.00 0.819605
\(317\) 4572.00i 0.810060i 0.914303 + 0.405030i \(0.132739\pi\)
−0.914303 + 0.405030i \(0.867261\pi\)
\(318\) 0 0
\(319\) −5580.00 −0.979373
\(320\) 0 0
\(321\) 0 0
\(322\) 468.000i 0.0809957i
\(323\) − 588.000i − 0.101292i
\(324\) 0 0
\(325\) 0 0
\(326\) −1346.00 −0.228675
\(327\) 0 0
\(328\) 3024.00i 0.509062i
\(329\) 1872.00 0.313698
\(330\) 0 0
\(331\) 845.000 0.140318 0.0701592 0.997536i \(-0.477649\pi\)
0.0701592 + 0.997536i \(0.477649\pi\)
\(332\) − 408.000i − 0.0674455i
\(333\) 0 0
\(334\) 7392.00 1.21099
\(335\) 0 0
\(336\) 0 0
\(337\) − 8723.00i − 1.41001i −0.709204 0.705003i \(-0.750946\pi\)
0.709204 0.705003i \(-0.249054\pi\)
\(338\) 3048.00i 0.490501i
\(339\) 0 0
\(340\) 0 0
\(341\) 4800.00 0.762271
\(342\) 0 0
\(343\) 6721.00i 1.05802i
\(344\) 2144.00 0.336037
\(345\) 0 0
\(346\) 6264.00 0.973280
\(347\) − 9018.00i − 1.39513i −0.716519 0.697567i \(-0.754266\pi\)
0.716519 0.697567i \(-0.245734\pi\)
\(348\) 0 0
\(349\) −5759.00 −0.883301 −0.441651 0.897187i \(-0.645607\pi\)
−0.441651 + 0.897187i \(0.645607\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 960.000i 0.145364i
\(353\) − 5772.00i − 0.870291i −0.900360 0.435145i \(-0.856697\pi\)
0.900360 0.435145i \(-0.143303\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 3936.00 0.585976
\(357\) 0 0
\(358\) 1020.00i 0.150583i
\(359\) 2046.00 0.300790 0.150395 0.988626i \(-0.451945\pi\)
0.150395 + 0.988626i \(0.451945\pi\)
\(360\) 0 0
\(361\) −4458.00 −0.649949
\(362\) 2174.00i 0.315643i
\(363\) 0 0
\(364\) −3172.00 −0.456753
\(365\) 0 0
\(366\) 0 0
\(367\) 1069.00i 0.152047i 0.997106 + 0.0760236i \(0.0242224\pi\)
−0.997106 + 0.0760236i \(0.975778\pi\)
\(368\) 288.000i 0.0407963i
\(369\) 0 0
\(370\) 0 0
\(371\) −7410.00 −1.03695
\(372\) 0 0
\(373\) 7133.00i 0.990168i 0.868845 + 0.495084i \(0.164863\pi\)
−0.868845 + 0.495084i \(0.835137\pi\)
\(374\) 720.000 0.0995463
\(375\) 0 0
\(376\) 1152.00 0.158005
\(377\) − 11346.0i − 1.55000i
\(378\) 0 0
\(379\) 8557.00 1.15975 0.579873 0.814707i \(-0.303102\pi\)
0.579873 + 0.814707i \(0.303102\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 8112.00i 1.08651i
\(383\) 14328.0i 1.91156i 0.294086 + 0.955779i \(0.404985\pi\)
−0.294086 + 0.955779i \(0.595015\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 946.000 0.124741
\(387\) 0 0
\(388\) − 1060.00i − 0.138694i
\(389\) −13500.0 −1.75958 −0.879791 0.475361i \(-0.842317\pi\)
−0.879791 + 0.475361i \(0.842317\pi\)
\(390\) 0 0
\(391\) 216.000 0.0279376
\(392\) 1392.00i 0.179354i
\(393\) 0 0
\(394\) −5112.00 −0.653652
\(395\) 0 0
\(396\) 0 0
\(397\) − 1334.00i − 0.168644i −0.996439 0.0843218i \(-0.973128\pi\)
0.996439 0.0843218i \(-0.0268724\pi\)
\(398\) − 5846.00i − 0.736265i
\(399\) 0 0
\(400\) 0 0
\(401\) 3474.00 0.432627 0.216313 0.976324i \(-0.430597\pi\)
0.216313 + 0.976324i \(0.430597\pi\)
\(402\) 0 0
\(403\) 9760.00i 1.20640i
\(404\) 4992.00 0.614756
\(405\) 0 0
\(406\) 4836.00 0.591149
\(407\) − 2730.00i − 0.332484i
\(408\) 0 0
\(409\) −569.000 −0.0687903 −0.0343952 0.999408i \(-0.510950\pi\)
−0.0343952 + 0.999408i \(0.510950\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 4900.00i 0.585936i
\(413\) − 2652.00i − 0.315972i
\(414\) 0 0
\(415\) 0 0
\(416\) −1952.00 −0.230059
\(417\) 0 0
\(418\) 2940.00i 0.344019i
\(419\) 9132.00 1.06474 0.532372 0.846511i \(-0.321301\pi\)
0.532372 + 0.846511i \(0.321301\pi\)
\(420\) 0 0
\(421\) −2971.00 −0.343937 −0.171969 0.985102i \(-0.555013\pi\)
−0.171969 + 0.985102i \(0.555013\pi\)
\(422\) 6350.00i 0.732496i
\(423\) 0 0
\(424\) −4560.00 −0.522295
\(425\) 0 0
\(426\) 0 0
\(427\) − 11401.0i − 1.29211i
\(428\) − 312.000i − 0.0352362i
\(429\) 0 0
\(430\) 0 0
\(431\) 12042.0 1.34581 0.672903 0.739730i \(-0.265047\pi\)
0.672903 + 0.739730i \(0.265047\pi\)
\(432\) 0 0
\(433\) − 8566.00i − 0.950706i −0.879795 0.475353i \(-0.842320\pi\)
0.879795 0.475353i \(-0.157680\pi\)
\(434\) −4160.00 −0.460107
\(435\) 0 0
\(436\) 8792.00 0.965735
\(437\) 882.000i 0.0965487i
\(438\) 0 0
\(439\) −7400.00 −0.804516 −0.402258 0.915526i \(-0.631775\pi\)
−0.402258 + 0.915526i \(0.631775\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1464.00i 0.157546i
\(443\) − 2580.00i − 0.276703i −0.990383 0.138352i \(-0.955820\pi\)
0.990383 0.138352i \(-0.0441804\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −4352.00 −0.462047
\(447\) 0 0
\(448\) − 832.000i − 0.0877418i
\(449\) −13200.0 −1.38741 −0.693704 0.720260i \(-0.744023\pi\)
−0.693704 + 0.720260i \(0.744023\pi\)
\(450\) 0 0
\(451\) −11340.0 −1.18399
\(452\) − 7944.00i − 0.826669i
\(453\) 0 0
\(454\) 7668.00 0.792681
\(455\) 0 0
\(456\) 0 0
\(457\) − 18038.0i − 1.84635i −0.384380 0.923175i \(-0.625585\pi\)
0.384380 0.923175i \(-0.374415\pi\)
\(458\) − 6404.00i − 0.653361i
\(459\) 0 0
\(460\) 0 0
\(461\) 5544.00 0.560108 0.280054 0.959984i \(-0.409648\pi\)
0.280054 + 0.959984i \(0.409648\pi\)
\(462\) 0 0
\(463\) − 17137.0i − 1.72014i −0.510178 0.860069i \(-0.670420\pi\)
0.510178 0.860069i \(-0.329580\pi\)
\(464\) 2976.00 0.297753
\(465\) 0 0
\(466\) 8304.00 0.825484
\(467\) 15888.0i 1.57432i 0.616747 + 0.787162i \(0.288450\pi\)
−0.616747 + 0.787162i \(0.711550\pi\)
\(468\) 0 0
\(469\) −2431.00 −0.239346
\(470\) 0 0
\(471\) 0 0
\(472\) − 1632.00i − 0.159150i
\(473\) 8040.00i 0.781564i
\(474\) 0 0
\(475\) 0 0
\(476\) −624.000 −0.0600861
\(477\) 0 0
\(478\) − 10932.0i − 1.04606i
\(479\) −3942.00 −0.376022 −0.188011 0.982167i \(-0.560204\pi\)
−0.188011 + 0.982167i \(0.560204\pi\)
\(480\) 0 0
\(481\) 5551.00 0.526203
\(482\) 1886.00i 0.178226i
\(483\) 0 0
\(484\) 1724.00 0.161908
\(485\) 0 0
\(486\) 0 0
\(487\) − 1379.00i − 0.128313i −0.997940 0.0641565i \(-0.979564\pi\)
0.997940 0.0641565i \(-0.0204357\pi\)
\(488\) − 7016.00i − 0.650818i
\(489\) 0 0
\(490\) 0 0
\(491\) 14214.0 1.30645 0.653227 0.757162i \(-0.273415\pi\)
0.653227 + 0.757162i \(0.273415\pi\)
\(492\) 0 0
\(493\) − 2232.00i − 0.203903i
\(494\) −5978.00 −0.544459
\(495\) 0 0
\(496\) −2560.00 −0.231749
\(497\) − 7878.00i − 0.711019i
\(498\) 0 0
\(499\) −9992.00 −0.896400 −0.448200 0.893933i \(-0.647935\pi\)
−0.448200 + 0.893933i \(0.647935\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 14580.0i 1.29629i
\(503\) − 21258.0i − 1.88439i −0.335067 0.942194i \(-0.608759\pi\)
0.335067 0.942194i \(-0.391241\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −1080.00 −0.0948851
\(507\) 0 0
\(508\) 11168.0i 0.975393i
\(509\) 16614.0 1.44676 0.723382 0.690448i \(-0.242587\pi\)
0.723382 + 0.690448i \(0.242587\pi\)
\(510\) 0 0
\(511\) −5603.00 −0.485053
\(512\) − 512.000i − 0.0441942i
\(513\) 0 0
\(514\) −624.000 −0.0535476
\(515\) 0 0
\(516\) 0 0
\(517\) 4320.00i 0.367492i
\(518\) 2366.00i 0.200687i
\(519\) 0 0
\(520\) 0 0
\(521\) 11838.0 0.995455 0.497728 0.867333i \(-0.334168\pi\)
0.497728 + 0.867333i \(0.334168\pi\)
\(522\) 0 0
\(523\) 2201.00i 0.184021i 0.995758 + 0.0920105i \(0.0293293\pi\)
−0.995758 + 0.0920105i \(0.970671\pi\)
\(524\) 2832.00 0.236100
\(525\) 0 0
\(526\) −16008.0 −1.32696
\(527\) 1920.00i 0.158703i
\(528\) 0 0
\(529\) 11843.0 0.973371
\(530\) 0 0
\(531\) 0 0
\(532\) − 2548.00i − 0.207650i
\(533\) − 23058.0i − 1.87383i
\(534\) 0 0
\(535\) 0 0
\(536\) −1496.00 −0.120555
\(537\) 0 0
\(538\) − 648.000i − 0.0519280i
\(539\) −5220.00 −0.417145
\(540\) 0 0
\(541\) −10795.0 −0.857880 −0.428940 0.903333i \(-0.641113\pi\)
−0.428940 + 0.903333i \(0.641113\pi\)
\(542\) 15698.0i 1.24407i
\(543\) 0 0
\(544\) −384.000 −0.0302645
\(545\) 0 0
\(546\) 0 0
\(547\) 14185.0i 1.10879i 0.832254 + 0.554394i \(0.187050\pi\)
−0.832254 + 0.554394i \(0.812950\pi\)
\(548\) − 6744.00i − 0.525711i
\(549\) 0 0
\(550\) 0 0
\(551\) 9114.00 0.704663
\(552\) 0 0
\(553\) − 14963.0i − 1.15062i
\(554\) 11516.0 0.883155
\(555\) 0 0
\(556\) −1228.00 −0.0936669
\(557\) 3576.00i 0.272029i 0.990707 + 0.136014i \(0.0434293\pi\)
−0.990707 + 0.136014i \(0.956571\pi\)
\(558\) 0 0
\(559\) −16348.0 −1.23694
\(560\) 0 0
\(561\) 0 0
\(562\) 5376.00i 0.403510i
\(563\) − 9132.00i − 0.683602i −0.939772 0.341801i \(-0.888963\pi\)
0.939772 0.341801i \(-0.111037\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 6520.00 0.484198
\(567\) 0 0
\(568\) − 4848.00i − 0.358130i
\(569\) −25020.0 −1.84340 −0.921699 0.387907i \(-0.873198\pi\)
−0.921699 + 0.387907i \(0.873198\pi\)
\(570\) 0 0
\(571\) −15997.0 −1.17242 −0.586212 0.810158i \(-0.699381\pi\)
−0.586212 + 0.810158i \(0.699381\pi\)
\(572\) − 7320.00i − 0.535078i
\(573\) 0 0
\(574\) 9828.00 0.714656
\(575\) 0 0
\(576\) 0 0
\(577\) 5971.00i 0.430808i 0.976525 + 0.215404i \(0.0691068\pi\)
−0.976525 + 0.215404i \(0.930893\pi\)
\(578\) − 9538.00i − 0.686381i
\(579\) 0 0
\(580\) 0 0
\(581\) −1326.00 −0.0946846
\(582\) 0 0
\(583\) − 17100.0i − 1.21477i
\(584\) −3448.00 −0.244314
\(585\) 0 0
\(586\) 11844.0 0.834934
\(587\) − 19242.0i − 1.35299i −0.736449 0.676493i \(-0.763499\pi\)
0.736449 0.676493i \(-0.236501\pi\)
\(588\) 0 0
\(589\) −7840.00 −0.548458
\(590\) 0 0
\(591\) 0 0
\(592\) 1456.00i 0.101083i
\(593\) 8118.00i 0.562169i 0.959683 + 0.281085i \(0.0906941\pi\)
−0.959683 + 0.281085i \(0.909306\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −7248.00 −0.498137
\(597\) 0 0
\(598\) − 2196.00i − 0.150169i
\(599\) −1902.00 −0.129739 −0.0648695 0.997894i \(-0.520663\pi\)
−0.0648695 + 0.997894i \(0.520663\pi\)
\(600\) 0 0
\(601\) −14074.0 −0.955225 −0.477613 0.878571i \(-0.658498\pi\)
−0.477613 + 0.878571i \(0.658498\pi\)
\(602\) − 6968.00i − 0.471752i
\(603\) 0 0
\(604\) −812.000 −0.0547017
\(605\) 0 0
\(606\) 0 0
\(607\) 13825.0i 0.924447i 0.886763 + 0.462224i \(0.152948\pi\)
−0.886763 + 0.462224i \(0.847052\pi\)
\(608\) − 1568.00i − 0.104590i
\(609\) 0 0
\(610\) 0 0
\(611\) −8784.00 −0.581608
\(612\) 0 0
\(613\) 15569.0i 1.02582i 0.858443 + 0.512909i \(0.171432\pi\)
−0.858443 + 0.512909i \(0.828568\pi\)
\(614\) −7456.00 −0.490065
\(615\) 0 0
\(616\) 3120.00 0.204072
\(617\) − 11922.0i − 0.777896i −0.921260 0.388948i \(-0.872839\pi\)
0.921260 0.388948i \(-0.127161\pi\)
\(618\) 0 0
\(619\) −6899.00 −0.447971 −0.223986 0.974592i \(-0.571907\pi\)
−0.223986 + 0.974592i \(0.571907\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 1464.00i 0.0943747i
\(623\) − 12792.0i − 0.822633i
\(624\) 0 0
\(625\) 0 0
\(626\) 10714.0 0.684054
\(627\) 0 0
\(628\) − 856.000i − 0.0543919i
\(629\) 1092.00 0.0692224
\(630\) 0 0
\(631\) 11711.0 0.738839 0.369420 0.929263i \(-0.379557\pi\)
0.369420 + 0.929263i \(0.379557\pi\)
\(632\) − 9208.00i − 0.579548i
\(633\) 0 0
\(634\) 9144.00 0.572799
\(635\) 0 0
\(636\) 0 0
\(637\) − 10614.0i − 0.660192i
\(638\) 11160.0i 0.692521i
\(639\) 0 0
\(640\) 0 0
\(641\) 9240.00 0.569357 0.284679 0.958623i \(-0.408113\pi\)
0.284679 + 0.958623i \(0.408113\pi\)
\(642\) 0 0
\(643\) − 17908.0i − 1.09832i −0.835716 0.549162i \(-0.814947\pi\)
0.835716 0.549162i \(-0.185053\pi\)
\(644\) 936.000 0.0572726
\(645\) 0 0
\(646\) −1176.00 −0.0716240
\(647\) − 7530.00i − 0.457550i −0.973479 0.228775i \(-0.926528\pi\)
0.973479 0.228775i \(-0.0734720\pi\)
\(648\) 0 0
\(649\) 6120.00 0.370156
\(650\) 0 0
\(651\) 0 0
\(652\) 2692.00i 0.161698i
\(653\) − 22788.0i − 1.36564i −0.730586 0.682820i \(-0.760753\pi\)
0.730586 0.682820i \(-0.239247\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 6048.00 0.359961
\(657\) 0 0
\(658\) − 3744.00i − 0.221818i
\(659\) −16440.0 −0.971793 −0.485896 0.874016i \(-0.661507\pi\)
−0.485896 + 0.874016i \(0.661507\pi\)
\(660\) 0 0
\(661\) −12421.0 −0.730894 −0.365447 0.930832i \(-0.619084\pi\)
−0.365447 + 0.930832i \(0.619084\pi\)
\(662\) − 1690.00i − 0.0992201i
\(663\) 0 0
\(664\) −816.000 −0.0476912
\(665\) 0 0
\(666\) 0 0
\(667\) 3348.00i 0.194355i
\(668\) − 14784.0i − 0.856303i
\(669\) 0 0
\(670\) 0 0
\(671\) 26310.0 1.51369
\(672\) 0 0
\(673\) 6461.00i 0.370064i 0.982732 + 0.185032i \(0.0592389\pi\)
−0.982732 + 0.185032i \(0.940761\pi\)
\(674\) −17446.0 −0.997025
\(675\) 0 0
\(676\) 6096.00 0.346837
\(677\) 912.000i 0.0517740i 0.999665 + 0.0258870i \(0.00824101\pi\)
−0.999665 + 0.0258870i \(0.991759\pi\)
\(678\) 0 0
\(679\) −3445.00 −0.194708
\(680\) 0 0
\(681\) 0 0
\(682\) − 9600.00i − 0.539007i
\(683\) − 14442.0i − 0.809089i −0.914518 0.404544i \(-0.867430\pi\)
0.914518 0.404544i \(-0.132570\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 13442.0 0.748131
\(687\) 0 0
\(688\) − 4288.00i − 0.237614i
\(689\) 34770.0 1.92254
\(690\) 0 0
\(691\) 1892.00 0.104161 0.0520804 0.998643i \(-0.483415\pi\)
0.0520804 + 0.998643i \(0.483415\pi\)
\(692\) − 12528.0i − 0.688213i
\(693\) 0 0
\(694\) −18036.0 −0.986509
\(695\) 0 0
\(696\) 0 0
\(697\) − 4536.00i − 0.246504i
\(698\) 11518.0i 0.624588i
\(699\) 0 0
\(700\) 0 0
\(701\) −7914.00 −0.426402 −0.213201 0.977008i \(-0.568389\pi\)
−0.213201 + 0.977008i \(0.568389\pi\)
\(702\) 0 0
\(703\) 4459.00i 0.239224i
\(704\) 1920.00 0.102788
\(705\) 0 0
\(706\) −11544.0 −0.615388
\(707\) − 16224.0i − 0.863036i
\(708\) 0 0
\(709\) 1291.00 0.0683844 0.0341922 0.999415i \(-0.489114\pi\)
0.0341922 + 0.999415i \(0.489114\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 7872.00i − 0.414348i
\(713\) − 2880.00i − 0.151272i
\(714\) 0 0
\(715\) 0 0
\(716\) 2040.00 0.106478
\(717\) 0 0
\(718\) − 4092.00i − 0.212691i
\(719\) 30210.0 1.56696 0.783479 0.621418i \(-0.213443\pi\)
0.783479 + 0.621418i \(0.213443\pi\)
\(720\) 0 0
\(721\) 15925.0 0.822577
\(722\) 8916.00i 0.459583i
\(723\) 0 0
\(724\) 4348.00 0.223194
\(725\) 0 0
\(726\) 0 0
\(727\) − 15680.0i − 0.799916i −0.916533 0.399958i \(-0.869025\pi\)
0.916533 0.399958i \(-0.130975\pi\)
\(728\) 6344.00i 0.322973i
\(729\) 0 0
\(730\) 0 0
\(731\) −3216.00 −0.162720
\(732\) 0 0
\(733\) 13898.0i 0.700320i 0.936690 + 0.350160i \(0.113873\pi\)
−0.936690 + 0.350160i \(0.886127\pi\)
\(734\) 2138.00 0.107514
\(735\) 0 0
\(736\) 576.000 0.0288473
\(737\) − 5610.00i − 0.280389i
\(738\) 0 0
\(739\) −35300.0 −1.75715 −0.878573 0.477607i \(-0.841504\pi\)
−0.878573 + 0.477607i \(0.841504\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 14820.0i 0.733234i
\(743\) 28188.0i 1.39181i 0.718132 + 0.695907i \(0.244997\pi\)
−0.718132 + 0.695907i \(0.755003\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 14266.0 0.700155
\(747\) 0 0
\(748\) − 1440.00i − 0.0703899i
\(749\) −1014.00 −0.0494670
\(750\) 0 0
\(751\) 25163.0 1.22265 0.611326 0.791379i \(-0.290637\pi\)
0.611326 + 0.791379i \(0.290637\pi\)
\(752\) − 2304.00i − 0.111726i
\(753\) 0 0
\(754\) −22692.0 −1.09601
\(755\) 0 0
\(756\) 0 0
\(757\) − 7979.00i − 0.383093i −0.981484 0.191547i \(-0.938650\pi\)
0.981484 0.191547i \(-0.0613503\pi\)
\(758\) − 17114.0i − 0.820064i
\(759\) 0 0
\(760\) 0 0
\(761\) 26622.0 1.26813 0.634065 0.773280i \(-0.281385\pi\)
0.634065 + 0.773280i \(0.281385\pi\)
\(762\) 0 0
\(763\) − 28574.0i − 1.35576i
\(764\) 16224.0 0.768277
\(765\) 0 0
\(766\) 28656.0 1.35168
\(767\) 12444.0i 0.585824i
\(768\) 0 0
\(769\) 35413.0 1.66063 0.830316 0.557293i \(-0.188160\pi\)
0.830316 + 0.557293i \(0.188160\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 1892.00i − 0.0882054i
\(773\) − 33834.0i − 1.57429i −0.616769 0.787144i \(-0.711559\pi\)
0.616769 0.787144i \(-0.288441\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −2120.00 −0.0980716
\(777\) 0 0
\(778\) 27000.0i 1.24421i
\(779\) 18522.0 0.851886
\(780\) 0 0
\(781\) 18180.0 0.832947
\(782\) − 432.000i − 0.0197548i
\(783\) 0 0
\(784\) 2784.00 0.126822
\(785\) 0 0
\(786\) 0 0
\(787\) − 13469.0i − 0.610061i −0.952343 0.305030i \(-0.901333\pi\)
0.952343 0.305030i \(-0.0986667\pi\)
\(788\) 10224.0i 0.462202i
\(789\) 0 0
\(790\) 0 0
\(791\) −25818.0 −1.16053
\(792\) 0 0
\(793\) 53497.0i 2.39563i
\(794\) −2668.00 −0.119249
\(795\) 0 0
\(796\) −11692.0 −0.520618
\(797\) 21168.0i 0.940789i 0.882456 + 0.470395i \(0.155888\pi\)
−0.882456 + 0.470395i \(0.844112\pi\)
\(798\) 0 0
\(799\) −1728.00 −0.0765109
\(800\) 0 0
\(801\) 0 0
\(802\) − 6948.00i − 0.305913i
\(803\) − 12930.0i − 0.568231i
\(804\) 0 0
\(805\) 0 0
\(806\) 19520.0 0.853055
\(807\) 0 0
\(808\) − 9984.00i − 0.434698i
\(809\) −408.000 −0.0177312 −0.00886558 0.999961i \(-0.502822\pi\)
−0.00886558 + 0.999961i \(0.502822\pi\)
\(810\) 0 0
\(811\) −36916.0 −1.59839 −0.799196 0.601070i \(-0.794741\pi\)
−0.799196 + 0.601070i \(0.794741\pi\)
\(812\) − 9672.00i − 0.418006i
\(813\) 0 0
\(814\) −5460.00 −0.235102
\(815\) 0 0
\(816\) 0 0
\(817\) − 13132.0i − 0.562338i
\(818\) 1138.00i 0.0486421i
\(819\) 0 0
\(820\) 0 0
\(821\) −24636.0 −1.04726 −0.523631 0.851945i \(-0.675423\pi\)
−0.523631 + 0.851945i \(0.675423\pi\)
\(822\) 0 0
\(823\) − 18187.0i − 0.770303i −0.922853 0.385151i \(-0.874149\pi\)
0.922853 0.385151i \(-0.125851\pi\)
\(824\) 9800.00 0.414319
\(825\) 0 0
\(826\) −5304.00 −0.223426
\(827\) − 1464.00i − 0.0615578i −0.999526 0.0307789i \(-0.990201\pi\)
0.999526 0.0307789i \(-0.00979877\pi\)
\(828\) 0 0
\(829\) 12295.0 0.515106 0.257553 0.966264i \(-0.417084\pi\)
0.257553 + 0.966264i \(0.417084\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 3904.00i 0.162676i
\(833\) − 2088.00i − 0.0868486i
\(834\) 0 0
\(835\) 0 0
\(836\) 5880.00 0.243258
\(837\) 0 0
\(838\) − 18264.0i − 0.752887i
\(839\) −46884.0 −1.92922 −0.964610 0.263681i \(-0.915063\pi\)
−0.964610 + 0.263681i \(0.915063\pi\)
\(840\) 0 0
\(841\) 10207.0 0.418508
\(842\) 5942.00i 0.243201i
\(843\) 0 0
\(844\) 12700.0 0.517953
\(845\) 0 0
\(846\) 0 0
\(847\) − 5603.00i − 0.227298i
\(848\) 9120.00i 0.369318i
\(849\) 0 0
\(850\) 0 0
\(851\) −1638.00 −0.0659811
\(852\) 0 0
\(853\) 12197.0i 0.489587i 0.969575 + 0.244793i \(0.0787201\pi\)
−0.969575 + 0.244793i \(0.921280\pi\)
\(854\) −22802.0 −0.913663
\(855\) 0 0
\(856\) −624.000 −0.0249157
\(857\) 28182.0i 1.12331i 0.827371 + 0.561656i \(0.189836\pi\)
−0.827371 + 0.561656i \(0.810164\pi\)
\(858\) 0 0
\(859\) −25433.0 −1.01020 −0.505101 0.863060i \(-0.668545\pi\)
−0.505101 + 0.863060i \(0.668545\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 24084.0i − 0.951629i
\(863\) − 16968.0i − 0.669290i −0.942344 0.334645i \(-0.891384\pi\)
0.942344 0.334645i \(-0.108616\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −17132.0 −0.672251
\(867\) 0 0
\(868\) 8320.00i 0.325345i
\(869\) 34530.0 1.34793
\(870\) 0 0
\(871\) 11407.0 0.443756
\(872\) − 17584.0i − 0.682878i
\(873\) 0 0
\(874\) 1764.00 0.0682702
\(875\) 0 0
\(876\) 0 0
\(877\) 22423.0i 0.863365i 0.902026 + 0.431682i \(0.142080\pi\)
−0.902026 + 0.431682i \(0.857920\pi\)
\(878\) 14800.0i 0.568879i
\(879\) 0 0
\(880\) 0 0
\(881\) 8442.00 0.322836 0.161418 0.986886i \(-0.448393\pi\)
0.161418 + 0.986886i \(0.448393\pi\)
\(882\) 0 0
\(883\) 41207.0i 1.57047i 0.619197 + 0.785236i \(0.287458\pi\)
−0.619197 + 0.785236i \(0.712542\pi\)
\(884\) 2928.00 0.111402
\(885\) 0 0
\(886\) −5160.00 −0.195659
\(887\) − 8472.00i − 0.320701i −0.987060 0.160351i \(-0.948738\pi\)
0.987060 0.160351i \(-0.0512625\pi\)
\(888\) 0 0
\(889\) 36296.0 1.36932
\(890\) 0 0
\(891\) 0 0
\(892\) 8704.00i 0.326717i
\(893\) − 7056.00i − 0.264412i
\(894\) 0 0
\(895\) 0 0
\(896\) −1664.00 −0.0620428
\(897\) 0 0
\(898\) 26400.0i 0.981046i
\(899\) −29760.0 −1.10406
\(900\) 0 0
\(901\) 6840.00 0.252912
\(902\) 22680.0i 0.837208i
\(903\) 0 0
\(904\) −15888.0 −0.584543
\(905\) 0 0
\(906\) 0 0
\(907\) 21799.0i 0.798042i 0.916942 + 0.399021i \(0.130650\pi\)
−0.916942 + 0.399021i \(0.869350\pi\)
\(908\) − 15336.0i − 0.560510i
\(909\) 0 0
\(910\) 0 0
\(911\) 23544.0 0.856254 0.428127 0.903719i \(-0.359174\pi\)
0.428127 + 0.903719i \(0.359174\pi\)
\(912\) 0 0
\(913\) − 3060.00i − 0.110921i
\(914\) −36076.0 −1.30557
\(915\) 0 0
\(916\) −12808.0 −0.461996
\(917\) − 9204.00i − 0.331453i
\(918\) 0 0
\(919\) −11072.0 −0.397423 −0.198711 0.980058i \(-0.563676\pi\)
−0.198711 + 0.980058i \(0.563676\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 11088.0i − 0.396056i
\(923\) 36966.0i 1.31826i
\(924\) 0 0
\(925\) 0 0
\(926\) −34274.0 −1.21632
\(927\) 0 0
\(928\) − 5952.00i − 0.210543i
\(929\) 21654.0 0.764741 0.382371 0.924009i \(-0.375108\pi\)
0.382371 + 0.924009i \(0.375108\pi\)
\(930\) 0 0
\(931\) 8526.00 0.300138
\(932\) − 16608.0i − 0.583705i
\(933\) 0 0
\(934\) 31776.0 1.11321
\(935\) 0 0
\(936\) 0 0
\(937\) 42835.0i 1.49345i 0.665135 + 0.746723i \(0.268374\pi\)
−0.665135 + 0.746723i \(0.731626\pi\)
\(938\) 4862.00i 0.169243i
\(939\) 0 0
\(940\) 0 0
\(941\) −48534.0 −1.68136 −0.840682 0.541529i \(-0.817845\pi\)
−0.840682 + 0.541529i \(0.817845\pi\)
\(942\) 0 0
\(943\) 6804.00i 0.234962i
\(944\) −3264.00 −0.112536
\(945\) 0 0
\(946\) 16080.0 0.552649
\(947\) − 14676.0i − 0.503597i −0.967780 0.251798i \(-0.918978\pi\)
0.967780 0.251798i \(-0.0810219\pi\)
\(948\) 0 0
\(949\) 26291.0 0.899307
\(950\) 0 0
\(951\) 0 0
\(952\) 1248.00i 0.0424873i
\(953\) 27372.0i 0.930395i 0.885207 + 0.465197i \(0.154017\pi\)
−0.885207 + 0.465197i \(0.845983\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −21864.0 −0.739678
\(957\) 0 0
\(958\) 7884.00i 0.265888i
\(959\) −21918.0 −0.738028
\(960\) 0 0
\(961\) −4191.00 −0.140680
\(962\) − 11102.0i − 0.372082i
\(963\) 0 0
\(964\) 3772.00 0.126025
\(965\) 0 0
\(966\) 0 0
\(967\) − 3581.00i − 0.119087i −0.998226 0.0595435i \(-0.981035\pi\)
0.998226 0.0595435i \(-0.0189645\pi\)
\(968\) − 3448.00i − 0.114486i
\(969\) 0 0
\(970\) 0 0
\(971\) −1824.00 −0.0602832 −0.0301416 0.999546i \(-0.509596\pi\)
−0.0301416 + 0.999546i \(0.509596\pi\)
\(972\) 0 0
\(973\) 3991.00i 0.131496i
\(974\) −2758.00 −0.0907310
\(975\) 0 0
\(976\) −14032.0 −0.460198
\(977\) 29778.0i 0.975110i 0.873092 + 0.487555i \(0.162111\pi\)
−0.873092 + 0.487555i \(0.837889\pi\)
\(978\) 0 0
\(979\) 29520.0 0.963701
\(980\) 0 0
\(981\) 0 0
\(982\) − 28428.0i − 0.923802i
\(983\) 9402.00i 0.305063i 0.988299 + 0.152532i \(0.0487426\pi\)
−0.988299 + 0.152532i \(0.951257\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −4464.00 −0.144181
\(987\) 0 0
\(988\) 11956.0i 0.384991i
\(989\) 4824.00 0.155100
\(990\) 0 0
\(991\) −24907.0 −0.798382 −0.399191 0.916868i \(-0.630709\pi\)
−0.399191 + 0.916868i \(0.630709\pi\)
\(992\) 5120.00i 0.163871i
\(993\) 0 0
\(994\) −15756.0 −0.502767
\(995\) 0 0
\(996\) 0 0
\(997\) − 27830.0i − 0.884037i −0.897006 0.442019i \(-0.854263\pi\)
0.897006 0.442019i \(-0.145737\pi\)
\(998\) 19984.0i 0.633850i
\(999\) 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 1350.4.c.f.649.1 2
3.2 odd 2 1350.4.c.o.649.2 2
5.2 odd 4 270.4.a.h.1.1 yes 1
5.3 odd 4 1350.4.a.i.1.1 1
5.4 even 2 inner 1350.4.c.f.649.2 2
15.2 even 4 270.4.a.d.1.1 1
15.8 even 4 1350.4.a.w.1.1 1
15.14 odd 2 1350.4.c.o.649.1 2
20.7 even 4 2160.4.a.g.1.1 1
45.2 even 12 810.4.e.r.271.1 2
45.7 odd 12 810.4.e.j.271.1 2
45.22 odd 12 810.4.e.j.541.1 2
45.32 even 12 810.4.e.r.541.1 2
60.47 odd 4 2160.4.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
270.4.a.d.1.1 1 15.2 even 4
270.4.a.h.1.1 yes 1 5.2 odd 4
810.4.e.j.271.1 2 45.7 odd 12
810.4.e.j.541.1 2 45.22 odd 12
810.4.e.r.271.1 2 45.2 even 12
810.4.e.r.541.1 2 45.32 even 12
1350.4.a.i.1.1 1 5.3 odd 4
1350.4.a.w.1.1 1 15.8 even 4
1350.4.c.f.649.1 2 1.1 even 1 trivial
1350.4.c.f.649.2 2 5.4 even 2 inner
1350.4.c.o.649.1 2 15.14 odd 2
1350.4.c.o.649.2 2 3.2 odd 2
2160.4.a.g.1.1 1 20.7 even 4
2160.4.a.q.1.1 1 60.47 odd 4