Properties

Label 3675.2.a.bq.1.4
Level $3675$
Weight $2$
Character 3675.1
Self dual yes
Analytic conductor $29.345$
Analytic rank $0$
Dimension $4$
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] = [3675,2,Mod(1,3675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3675, 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("3675.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3675.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: \(29.3450227428\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.88404.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 7x^{2} + 5x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 525)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-2.43736\) of defining polynomial
Character \(\chi\) \(=\) 3675.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.43736 q^{2} -1.00000 q^{3} +3.94072 q^{4} -2.43736 q^{6} +4.73024 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.43736 q^{2} -1.00000 q^{3} +3.94072 q^{4} -2.43736 q^{6} +4.73024 q^{8} +1.00000 q^{9} +4.58184 q^{11} -3.94072 q^{12} -1.35888 q^{13} +3.64784 q^{16} -4.29288 q^{17} +2.43736 q^{18} +6.81544 q^{19} +11.1676 q^{22} +3.88144 q^{23} -4.73024 q^{24} -3.31208 q^{26} -1.00000 q^{27} +4.00000 q^{29} +8.10832 q^{31} -0.569365 q^{32} -4.58184 q^{33} -10.4633 q^{34} +3.94072 q^{36} -11.7562 q^{37} +16.6117 q^{38} +1.35888 q^{39} -8.17432 q^{41} +3.52256 q^{43} +18.0558 q^{44} +9.46047 q^{46} +3.29960 q^{47} -3.64784 q^{48} +4.29288 q^{51} -5.35497 q^{52} +9.45656 q^{53} -2.43736 q^{54} -6.81544 q^{57} +9.74944 q^{58} -0.700398 q^{59} +5.00000 q^{61} +19.7629 q^{62} -8.68344 q^{64} -11.1676 q^{66} +1.94072 q^{67} -16.9170 q^{68} -3.88144 q^{69} +11.8680 q^{71} +4.73024 q^{72} +6.16368 q^{73} -28.6540 q^{74} +26.8578 q^{76} +3.31208 q^{78} -6.16760 q^{79} +1.00000 q^{81} -19.9238 q^{82} +13.0490 q^{83} +8.58575 q^{86} -4.00000 q^{87} +21.6732 q^{88} -0.292877 q^{89} +15.2957 q^{92} -8.10832 q^{93} +8.04232 q^{94} +0.569365 q^{96} -2.00672 q^{97} +4.58184 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - 4 q^{3} + 7 q^{4} + q^{6} - 3 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} - 4 q^{3} + 7 q^{4} + q^{6} - 3 q^{8} + 4 q^{9} + 8 q^{11} - 7 q^{12} - 7 q^{13} + 17 q^{16} - 6 q^{17} - q^{18} - 3 q^{19} + 12 q^{22} - 2 q^{23} + 3 q^{24} + 19 q^{26} - 4 q^{27} + 16 q^{29} - 9 q^{31} - 17 q^{32} - 8 q^{33} - 14 q^{34} + 7 q^{36} - 8 q^{37} + 27 q^{38} + 7 q^{39} - 4 q^{41} - 5 q^{43} + 26 q^{44} - 6 q^{46} + 6 q^{47} - 17 q^{48} + 6 q^{51} - 35 q^{52} + 6 q^{53} + q^{54} + 3 q^{57} - 4 q^{58} - 10 q^{59} + 20 q^{61} + 44 q^{62} + 21 q^{64} - 12 q^{66} - q^{67} + 8 q^{68} + 2 q^{69} + 22 q^{71} - 3 q^{72} + 4 q^{73} - 21 q^{74} + 23 q^{76} - 19 q^{78} + 8 q^{79} + 4 q^{81} - 8 q^{82} + 2 q^{83} + 12 q^{86} - 16 q^{87} - 28 q^{88} + 10 q^{89} + 66 q^{92} + 9 q^{93} - 22 q^{94} + 17 q^{96} - 12 q^{97} + 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.43736 1.72347 0.861737 0.507356i \(-0.169377\pi\)
0.861737 + 0.507356i \(0.169377\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.94072 1.97036
\(5\) 0 0
\(6\) −2.43736 −0.995048
\(7\) 0 0
\(8\) 4.73024 1.67239
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.58184 1.38148 0.690739 0.723104i \(-0.257286\pi\)
0.690739 + 0.723104i \(0.257286\pi\)
\(12\) −3.94072 −1.13759
\(13\) −1.35888 −0.376885 −0.188443 0.982084i \(-0.560344\pi\)
−0.188443 + 0.982084i \(0.560344\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 3.64784 0.911961
\(17\) −4.29288 −1.04118 −0.520588 0.853808i \(-0.674287\pi\)
−0.520588 + 0.853808i \(0.674287\pi\)
\(18\) 2.43736 0.574491
\(19\) 6.81544 1.56357 0.781785 0.623548i \(-0.214310\pi\)
0.781785 + 0.623548i \(0.214310\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 11.1676 2.38094
\(23\) 3.88144 0.809337 0.404668 0.914463i \(-0.367387\pi\)
0.404668 + 0.914463i \(0.367387\pi\)
\(24\) −4.73024 −0.965556
\(25\) 0 0
\(26\) −3.31208 −0.649552
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 8.10832 1.45630 0.728148 0.685419i \(-0.240381\pi\)
0.728148 + 0.685419i \(0.240381\pi\)
\(32\) −0.569365 −0.100650
\(33\) −4.58184 −0.797596
\(34\) −10.4633 −1.79444
\(35\) 0 0
\(36\) 3.94072 0.656787
\(37\) −11.7562 −1.93270 −0.966351 0.257228i \(-0.917191\pi\)
−0.966351 + 0.257228i \(0.917191\pi\)
\(38\) 16.6117 2.69477
\(39\) 1.35888 0.217595
\(40\) 0 0
\(41\) −8.17432 −1.27661 −0.638307 0.769782i \(-0.720365\pi\)
−0.638307 + 0.769782i \(0.720365\pi\)
\(42\) 0 0
\(43\) 3.52256 0.537186 0.268593 0.963254i \(-0.413441\pi\)
0.268593 + 0.963254i \(0.413441\pi\)
\(44\) 18.0558 2.72201
\(45\) 0 0
\(46\) 9.46047 1.39487
\(47\) 3.29960 0.481296 0.240648 0.970612i \(-0.422640\pi\)
0.240648 + 0.970612i \(0.422640\pi\)
\(48\) −3.64784 −0.526521
\(49\) 0 0
\(50\) 0 0
\(51\) 4.29288 0.601123
\(52\) −5.35497 −0.742600
\(53\) 9.45656 1.29896 0.649479 0.760379i \(-0.274987\pi\)
0.649479 + 0.760379i \(0.274987\pi\)
\(54\) −2.43736 −0.331683
\(55\) 0 0
\(56\) 0 0
\(57\) −6.81544 −0.902727
\(58\) 9.74944 1.28016
\(59\) −0.700398 −0.0911841 −0.0455921 0.998960i \(-0.514517\pi\)
−0.0455921 + 0.998960i \(0.514517\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 19.7629 2.50989
\(63\) 0 0
\(64\) −8.68344 −1.08543
\(65\) 0 0
\(66\) −11.1676 −1.37464
\(67\) 1.94072 0.237097 0.118548 0.992948i \(-0.462176\pi\)
0.118548 + 0.992948i \(0.462176\pi\)
\(68\) −16.9170 −2.05149
\(69\) −3.88144 −0.467271
\(70\) 0 0
\(71\) 11.8680 1.40847 0.704236 0.709966i \(-0.251290\pi\)
0.704236 + 0.709966i \(0.251290\pi\)
\(72\) 4.73024 0.557464
\(73\) 6.16368 0.721405 0.360702 0.932681i \(-0.382537\pi\)
0.360702 + 0.932681i \(0.382537\pi\)
\(74\) −28.6540 −3.33096
\(75\) 0 0
\(76\) 26.8578 3.08080
\(77\) 0 0
\(78\) 3.31208 0.375019
\(79\) −6.16760 −0.693909 −0.346954 0.937882i \(-0.612784\pi\)
−0.346954 + 0.937882i \(0.612784\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −19.9238 −2.20021
\(83\) 13.0490 1.43232 0.716159 0.697937i \(-0.245898\pi\)
0.716159 + 0.697937i \(0.245898\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.58575 0.925826
\(87\) −4.00000 −0.428845
\(88\) 21.6732 2.31037
\(89\) −0.292877 −0.0310449 −0.0155225 0.999880i \(-0.504941\pi\)
−0.0155225 + 0.999880i \(0.504941\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 15.2957 1.59469
\(93\) −8.10832 −0.840793
\(94\) 8.04232 0.829502
\(95\) 0 0
\(96\) 0.569365 0.0581105
\(97\) −2.00672 −0.203752 −0.101876 0.994797i \(-0.532484\pi\)
−0.101876 + 0.994797i \(0.532484\pi\)
\(98\) 0 0
\(99\) 4.58184 0.460492
\(100\) 0 0
\(101\) −6.29288 −0.626165 −0.313082 0.949726i \(-0.601362\pi\)
−0.313082 + 0.949726i \(0.601362\pi\)
\(102\) 10.4633 1.03602
\(103\) −3.41143 −0.336139 −0.168069 0.985775i \(-0.553753\pi\)
−0.168069 + 0.985775i \(0.553753\pi\)
\(104\) −6.42782 −0.630300
\(105\) 0 0
\(106\) 23.0490 2.23872
\(107\) −3.16760 −0.306223 −0.153112 0.988209i \(-0.548929\pi\)
−0.153112 + 0.988209i \(0.548929\pi\)
\(108\) −3.94072 −0.379196
\(109\) 10.2336 0.980201 0.490101 0.871666i \(-0.336960\pi\)
0.490101 + 0.871666i \(0.336960\pi\)
\(110\) 0 0
\(111\) 11.7562 1.11585
\(112\) 0 0
\(113\) 3.16368 0.297614 0.148807 0.988866i \(-0.452457\pi\)
0.148807 + 0.988866i \(0.452457\pi\)
\(114\) −16.6117 −1.55583
\(115\) 0 0
\(116\) 15.7629 1.46355
\(117\) −1.35888 −0.125628
\(118\) −1.70712 −0.157153
\(119\) 0 0
\(120\) 0 0
\(121\) 9.99328 0.908480
\(122\) 12.1868 1.10334
\(123\) 8.17432 0.737054
\(124\) 31.9526 2.86943
\(125\) 0 0
\(126\) 0 0
\(127\) −0.0106369 −0.000943870 0 −0.000471935 1.00000i \(-0.500150\pi\)
−0.000471935 1.00000i \(0.500150\pi\)
\(128\) −20.0259 −1.77006
\(129\) −3.52256 −0.310144
\(130\) 0 0
\(131\) −13.7629 −1.20247 −0.601235 0.799073i \(-0.705324\pi\)
−0.601235 + 0.799073i \(0.705324\pi\)
\(132\) −18.0558 −1.57155
\(133\) 0 0
\(134\) 4.73024 0.408630
\(135\) 0 0
\(136\) −20.3063 −1.74125
\(137\) −9.04513 −0.772777 −0.386389 0.922336i \(-0.626278\pi\)
−0.386389 + 0.922336i \(0.626278\pi\)
\(138\) −9.46047 −0.805329
\(139\) 21.3313 1.80930 0.904648 0.426160i \(-0.140134\pi\)
0.904648 + 0.426160i \(0.140134\pi\)
\(140\) 0 0
\(141\) −3.29960 −0.277877
\(142\) 28.9266 2.42746
\(143\) −6.22617 −0.520659
\(144\) 3.64784 0.303987
\(145\) 0 0
\(146\) 15.0231 1.24332
\(147\) 0 0
\(148\) −46.3278 −3.80812
\(149\) −15.8052 −1.29481 −0.647406 0.762145i \(-0.724146\pi\)
−0.647406 + 0.762145i \(0.724146\pi\)
\(150\) 0 0
\(151\) 4.86408 0.395833 0.197917 0.980219i \(-0.436582\pi\)
0.197917 + 0.980219i \(0.436582\pi\)
\(152\) 32.2386 2.61490
\(153\) −4.29288 −0.347059
\(154\) 0 0
\(155\) 0 0
\(156\) 5.35497 0.428741
\(157\) −6.57903 −0.525064 −0.262532 0.964923i \(-0.584557\pi\)
−0.262532 + 0.964923i \(0.584557\pi\)
\(158\) −15.0327 −1.19593
\(159\) −9.45656 −0.749954
\(160\) 0 0
\(161\) 0 0
\(162\) 2.43736 0.191497
\(163\) −11.4633 −0.897874 −0.448937 0.893563i \(-0.648197\pi\)
−0.448937 + 0.893563i \(0.648197\pi\)
\(164\) −32.2127 −2.51539
\(165\) 0 0
\(166\) 31.8052 2.46856
\(167\) 20.9170 1.61861 0.809304 0.587390i \(-0.199844\pi\)
0.809304 + 0.587390i \(0.199844\pi\)
\(168\) 0 0
\(169\) −11.1534 −0.857957
\(170\) 0 0
\(171\) 6.81544 0.521190
\(172\) 13.8814 1.05845
\(173\) −11.1637 −0.848759 −0.424380 0.905484i \(-0.639508\pi\)
−0.424380 + 0.905484i \(0.639508\pi\)
\(174\) −9.74944 −0.739103
\(175\) 0 0
\(176\) 16.7138 1.25985
\(177\) 0.700398 0.0526452
\(178\) −0.713847 −0.0535051
\(179\) −17.5123 −1.30893 −0.654466 0.756091i \(-0.727107\pi\)
−0.654466 + 0.756091i \(0.727107\pi\)
\(180\) 0 0
\(181\) −20.2653 −1.50631 −0.753153 0.657845i \(-0.771468\pi\)
−0.753153 + 0.657845i \(0.771468\pi\)
\(182\) 0 0
\(183\) −5.00000 −0.369611
\(184\) 18.3601 1.35353
\(185\) 0 0
\(186\) −19.7629 −1.44909
\(187\) −19.6693 −1.43836
\(188\) 13.0028 0.948328
\(189\) 0 0
\(190\) 0 0
\(191\) −4.13592 −0.299265 −0.149632 0.988742i \(-0.547809\pi\)
−0.149632 + 0.988742i \(0.547809\pi\)
\(192\) 8.68344 0.626673
\(193\) 4.76640 0.343093 0.171547 0.985176i \(-0.445124\pi\)
0.171547 + 0.985176i \(0.445124\pi\)
\(194\) −4.89111 −0.351161
\(195\) 0 0
\(196\) 0 0
\(197\) 5.30351 0.377860 0.188930 0.981991i \(-0.439498\pi\)
0.188930 + 0.981991i \(0.439498\pi\)
\(198\) 11.1676 0.793647
\(199\) 2.27833 0.161506 0.0807532 0.996734i \(-0.474267\pi\)
0.0807532 + 0.996734i \(0.474267\pi\)
\(200\) 0 0
\(201\) −1.94072 −0.136888
\(202\) −15.3380 −1.07918
\(203\) 0 0
\(204\) 16.9170 1.18443
\(205\) 0 0
\(206\) −8.31489 −0.579326
\(207\) 3.88144 0.269779
\(208\) −4.95698 −0.343705
\(209\) 31.2273 2.16004
\(210\) 0 0
\(211\) −14.8052 −1.01923 −0.509616 0.860402i \(-0.670213\pi\)
−0.509616 + 0.860402i \(0.670213\pi\)
\(212\) 37.2657 2.55942
\(213\) −11.8680 −0.813182
\(214\) −7.72057 −0.527767
\(215\) 0 0
\(216\) −4.73024 −0.321852
\(217\) 0 0
\(218\) 24.9430 1.68935
\(219\) −6.16368 −0.416503
\(220\) 0 0
\(221\) 5.83350 0.392404
\(222\) 28.6540 1.92313
\(223\) 6.63479 0.444299 0.222149 0.975013i \(-0.428693\pi\)
0.222149 + 0.975013i \(0.428693\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 7.71104 0.512930
\(227\) −9.03168 −0.599454 −0.299727 0.954025i \(-0.596896\pi\)
−0.299727 + 0.954025i \(0.596896\pi\)
\(228\) −26.8578 −1.77870
\(229\) −16.8328 −1.11234 −0.556171 0.831068i \(-0.687730\pi\)
−0.556171 + 0.831068i \(0.687730\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 18.9209 1.24222
\(233\) −18.2929 −1.19841 −0.599203 0.800597i \(-0.704516\pi\)
−0.599203 + 0.800597i \(0.704516\pi\)
\(234\) −3.31208 −0.216517
\(235\) 0 0
\(236\) −2.76008 −0.179666
\(237\) 6.16760 0.400628
\(238\) 0 0
\(239\) 11.6309 0.752339 0.376170 0.926551i \(-0.377241\pi\)
0.376170 + 0.926551i \(0.377241\pi\)
\(240\) 0 0
\(241\) 1.76961 0.113991 0.0569953 0.998374i \(-0.481848\pi\)
0.0569953 + 0.998374i \(0.481848\pi\)
\(242\) 24.3572 1.56574
\(243\) −1.00000 −0.0641500
\(244\) 19.7036 1.26139
\(245\) 0 0
\(246\) 19.9238 1.27029
\(247\) −9.26137 −0.589287
\(248\) 38.3543 2.43550
\(249\) −13.0490 −0.826949
\(250\) 0 0
\(251\) 10.7004 0.675403 0.337702 0.941253i \(-0.390351\pi\)
0.337702 + 0.941253i \(0.390351\pi\)
\(252\) 0 0
\(253\) 17.7842 1.11808
\(254\) −0.0259259 −0.00162674
\(255\) 0 0
\(256\) −31.4435 −1.96522
\(257\) 26.1609 1.63187 0.815935 0.578143i \(-0.196223\pi\)
0.815935 + 0.578143i \(0.196223\pi\)
\(258\) −8.58575 −0.534526
\(259\) 0 0
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) −33.5451 −2.07242
\(263\) −13.3803 −0.825066 −0.412533 0.910943i \(-0.635356\pi\)
−0.412533 + 0.910943i \(0.635356\pi\)
\(264\) −21.6732 −1.33389
\(265\) 0 0
\(266\) 0 0
\(267\) 0.292877 0.0179238
\(268\) 7.64784 0.467166
\(269\) 2.87081 0.175036 0.0875181 0.996163i \(-0.472106\pi\)
0.0875181 + 0.996163i \(0.472106\pi\)
\(270\) 0 0
\(271\) −29.3419 −1.78239 −0.891197 0.453616i \(-0.850134\pi\)
−0.891197 + 0.453616i \(0.850134\pi\)
\(272\) −15.6597 −0.949512
\(273\) 0 0
\(274\) −22.0462 −1.33186
\(275\) 0 0
\(276\) −15.2957 −0.920692
\(277\) −19.9830 −1.20066 −0.600332 0.799751i \(-0.704965\pi\)
−0.600332 + 0.799751i \(0.704965\pi\)
\(278\) 51.9920 3.11827
\(279\) 8.10832 0.485432
\(280\) 0 0
\(281\) 7.12919 0.425292 0.212646 0.977129i \(-0.431792\pi\)
0.212646 + 0.977129i \(0.431792\pi\)
\(282\) −8.04232 −0.478913
\(283\) 22.8751 1.35978 0.679892 0.733312i \(-0.262027\pi\)
0.679892 + 0.733312i \(0.262027\pi\)
\(284\) 46.7685 2.77520
\(285\) 0 0
\(286\) −15.1754 −0.897341
\(287\) 0 0
\(288\) −0.569365 −0.0335501
\(289\) 1.42879 0.0840468
\(290\) 0 0
\(291\) 2.00672 0.117636
\(292\) 24.2894 1.42143
\(293\) −3.63088 −0.212118 −0.106059 0.994360i \(-0.533823\pi\)
−0.106059 + 0.994360i \(0.533823\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −55.6094 −3.23223
\(297\) −4.58184 −0.265865
\(298\) −38.5230 −2.23158
\(299\) −5.27442 −0.305027
\(300\) 0 0
\(301\) 0 0
\(302\) 11.8555 0.682208
\(303\) 6.29288 0.361516
\(304\) 24.8617 1.42591
\(305\) 0 0
\(306\) −10.4633 −0.598146
\(307\) −6.22687 −0.355387 −0.177693 0.984086i \(-0.556863\pi\)
−0.177693 + 0.984086i \(0.556863\pi\)
\(308\) 0 0
\(309\) 3.41143 0.194070
\(310\) 0 0
\(311\) 21.6309 1.22657 0.613287 0.789860i \(-0.289847\pi\)
0.613287 + 0.789860i \(0.289847\pi\)
\(312\) 6.42782 0.363904
\(313\) 23.8962 1.35069 0.675345 0.737502i \(-0.263995\pi\)
0.675345 + 0.737502i \(0.263995\pi\)
\(314\) −16.0355 −0.904933
\(315\) 0 0
\(316\) −24.3048 −1.36725
\(317\) −8.57231 −0.481469 −0.240734 0.970591i \(-0.577388\pi\)
−0.240734 + 0.970591i \(0.577388\pi\)
\(318\) −23.0490 −1.29253
\(319\) 18.3274 1.02614
\(320\) 0 0
\(321\) 3.16760 0.176798
\(322\) 0 0
\(323\) −29.2579 −1.62795
\(324\) 3.94072 0.218929
\(325\) 0 0
\(326\) −27.9401 −1.54746
\(327\) −10.2336 −0.565919
\(328\) −38.6665 −2.13500
\(329\) 0 0
\(330\) 0 0
\(331\) 13.5095 0.742550 0.371275 0.928523i \(-0.378921\pi\)
0.371275 + 0.928523i \(0.378921\pi\)
\(332\) 51.4226 2.82218
\(333\) −11.7562 −0.644234
\(334\) 50.9823 2.78963
\(335\) 0 0
\(336\) 0 0
\(337\) −11.1016 −0.604742 −0.302371 0.953190i \(-0.597778\pi\)
−0.302371 + 0.953190i \(0.597778\pi\)
\(338\) −27.1850 −1.47867
\(339\) −3.16368 −0.171828
\(340\) 0 0
\(341\) 37.1510 2.01184
\(342\) 16.6117 0.898257
\(343\) 0 0
\(344\) 16.6626 0.898385
\(345\) 0 0
\(346\) −27.2099 −1.46281
\(347\) −27.3842 −1.47006 −0.735031 0.678033i \(-0.762833\pi\)
−0.735031 + 0.678033i \(0.762833\pi\)
\(348\) −15.7629 −0.844979
\(349\) −6.17823 −0.330713 −0.165357 0.986234i \(-0.552878\pi\)
−0.165357 + 0.986234i \(0.552878\pi\)
\(350\) 0 0
\(351\) 1.35888 0.0725316
\(352\) −2.60874 −0.139046
\(353\) −7.80520 −0.415429 −0.207715 0.978189i \(-0.566603\pi\)
−0.207715 + 0.978189i \(0.566603\pi\)
\(354\) 1.70712 0.0907326
\(355\) 0 0
\(356\) −1.15415 −0.0611697
\(357\) 0 0
\(358\) −42.6838 −2.25591
\(359\) 4.11464 0.217163 0.108581 0.994088i \(-0.465369\pi\)
0.108581 + 0.994088i \(0.465369\pi\)
\(360\) 0 0
\(361\) 27.4502 1.44475
\(362\) −49.3938 −2.59608
\(363\) −9.99328 −0.524511
\(364\) 0 0
\(365\) 0 0
\(366\) −12.1868 −0.637014
\(367\) −21.8115 −1.13855 −0.569276 0.822146i \(-0.692777\pi\)
−0.569276 + 0.822146i \(0.692777\pi\)
\(368\) 14.1589 0.738084
\(369\) −8.17432 −0.425538
\(370\) 0 0
\(371\) 0 0
\(372\) −31.9526 −1.65667
\(373\) −2.93791 −0.152119 −0.0760596 0.997103i \(-0.524234\pi\)
−0.0760596 + 0.997103i \(0.524234\pi\)
\(374\) −47.9411 −2.47898
\(375\) 0 0
\(376\) 15.6079 0.804916
\(377\) −5.43552 −0.279943
\(378\) 0 0
\(379\) −8.85384 −0.454791 −0.227396 0.973802i \(-0.573021\pi\)
−0.227396 + 0.973802i \(0.573021\pi\)
\(380\) 0 0
\(381\) 0.0106369 0.000544944 0
\(382\) −10.0807 −0.515774
\(383\) −17.6387 −0.901296 −0.450648 0.892702i \(-0.648807\pi\)
−0.450648 + 0.892702i \(0.648807\pi\)
\(384\) 20.0259 1.02194
\(385\) 0 0
\(386\) 11.6174 0.591312
\(387\) 3.52256 0.179062
\(388\) −7.90794 −0.401465
\(389\) 18.1878 0.922157 0.461078 0.887359i \(-0.347463\pi\)
0.461078 + 0.887359i \(0.347463\pi\)
\(390\) 0 0
\(391\) −16.6626 −0.842662
\(392\) 0 0
\(393\) 13.7629 0.694246
\(394\) 12.9266 0.651231
\(395\) 0 0
\(396\) 18.0558 0.907336
\(397\) −10.5237 −0.528168 −0.264084 0.964500i \(-0.585070\pi\)
−0.264084 + 0.964500i \(0.585070\pi\)
\(398\) 5.55310 0.278352
\(399\) 0 0
\(400\) 0 0
\(401\) 3.58857 0.179204 0.0896022 0.995978i \(-0.471440\pi\)
0.0896022 + 0.995978i \(0.471440\pi\)
\(402\) −4.73024 −0.235923
\(403\) −11.0182 −0.548857
\(404\) −24.7985 −1.23377
\(405\) 0 0
\(406\) 0 0
\(407\) −53.8649 −2.66998
\(408\) 20.3063 1.00531
\(409\) 22.5304 1.11406 0.557028 0.830494i \(-0.311942\pi\)
0.557028 + 0.830494i \(0.311942\pi\)
\(410\) 0 0
\(411\) 9.04513 0.446163
\(412\) −13.4435 −0.662314
\(413\) 0 0
\(414\) 9.46047 0.464957
\(415\) 0 0
\(416\) 0.773698 0.0379337
\(417\) −21.3313 −1.04460
\(418\) 76.1121 3.72276
\(419\) −39.4055 −1.92508 −0.962542 0.271131i \(-0.912602\pi\)
−0.962542 + 0.271131i \(0.912602\pi\)
\(420\) 0 0
\(421\) −8.57121 −0.417735 −0.208867 0.977944i \(-0.566978\pi\)
−0.208867 + 0.977944i \(0.566978\pi\)
\(422\) −36.0856 −1.75662
\(423\) 3.29960 0.160432
\(424\) 44.7318 2.17237
\(425\) 0 0
\(426\) −28.9266 −1.40150
\(427\) 0 0
\(428\) −12.4826 −0.603370
\(429\) 6.22617 0.300602
\(430\) 0 0
\(431\) 1.28224 0.0617633 0.0308817 0.999523i \(-0.490168\pi\)
0.0308817 + 0.999523i \(0.490168\pi\)
\(432\) −3.64784 −0.175507
\(433\) 5.21233 0.250488 0.125244 0.992126i \(-0.460029\pi\)
0.125244 + 0.992126i \(0.460029\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 40.3278 1.93135
\(437\) 26.4537 1.26545
\(438\) −15.0231 −0.717832
\(439\) −19.6281 −0.936797 −0.468398 0.883517i \(-0.655169\pi\)
−0.468398 + 0.883517i \(0.655169\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 14.2183 0.676298
\(443\) 18.0347 0.856856 0.428428 0.903576i \(-0.359068\pi\)
0.428428 + 0.903576i \(0.359068\pi\)
\(444\) 46.3278 2.19862
\(445\) 0 0
\(446\) 16.1714 0.765737
\(447\) 15.8052 0.747560
\(448\) 0 0
\(449\) −6.99719 −0.330218 −0.165109 0.986275i \(-0.552798\pi\)
−0.165109 + 0.986275i \(0.552798\pi\)
\(450\) 0 0
\(451\) −37.4534 −1.76361
\(452\) 12.4672 0.586408
\(453\) −4.86408 −0.228535
\(454\) −22.0134 −1.03314
\(455\) 0 0
\(456\) −32.2386 −1.50971
\(457\) −37.8408 −1.77012 −0.885059 0.465479i \(-0.845882\pi\)
−0.885059 + 0.465479i \(0.845882\pi\)
\(458\) −41.0276 −1.91709
\(459\) 4.29288 0.200374
\(460\) 0 0
\(461\) −24.0903 −1.12200 −0.560998 0.827817i \(-0.689582\pi\)
−0.560998 + 0.827817i \(0.689582\pi\)
\(462\) 0 0
\(463\) −23.1154 −1.07427 −0.537133 0.843498i \(-0.680493\pi\)
−0.537133 + 0.843498i \(0.680493\pi\)
\(464\) 14.5914 0.677388
\(465\) 0 0
\(466\) −44.5863 −2.06542
\(467\) 12.5645 0.581415 0.290707 0.956812i \(-0.406109\pi\)
0.290707 + 0.956812i \(0.406109\pi\)
\(468\) −5.35497 −0.247533
\(469\) 0 0
\(470\) 0 0
\(471\) 6.57903 0.303146
\(472\) −3.31305 −0.152496
\(473\) 16.1398 0.742110
\(474\) 15.0327 0.690472
\(475\) 0 0
\(476\) 0 0
\(477\) 9.45656 0.432986
\(478\) 28.3486 1.29664
\(479\) 2.03081 0.0927901 0.0463950 0.998923i \(-0.485227\pi\)
0.0463950 + 0.998923i \(0.485227\pi\)
\(480\) 0 0
\(481\) 15.9752 0.728407
\(482\) 4.31318 0.196460
\(483\) 0 0
\(484\) 39.3807 1.79003
\(485\) 0 0
\(486\) −2.43736 −0.110561
\(487\) −27.5226 −1.24717 −0.623583 0.781757i \(-0.714324\pi\)
−0.623583 + 0.781757i \(0.714324\pi\)
\(488\) 23.6512 1.07064
\(489\) 11.4633 0.518388
\(490\) 0 0
\(491\) 0.459373 0.0207312 0.0103656 0.999946i \(-0.496700\pi\)
0.0103656 + 0.999946i \(0.496700\pi\)
\(492\) 32.2127 1.45226
\(493\) −17.1715 −0.773366
\(494\) −22.5733 −1.01562
\(495\) 0 0
\(496\) 29.5779 1.32809
\(497\) 0 0
\(498\) −31.8052 −1.42523
\(499\) −11.6383 −0.521002 −0.260501 0.965474i \(-0.583888\pi\)
−0.260501 + 0.965474i \(0.583888\pi\)
\(500\) 0 0
\(501\) −20.9170 −0.934504
\(502\) 26.0807 1.16404
\(503\) −32.4254 −1.44578 −0.722890 0.690964i \(-0.757187\pi\)
−0.722890 + 0.690964i \(0.757187\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 43.3464 1.92698
\(507\) 11.1534 0.495342
\(508\) −0.0419170 −0.00185977
\(509\) −17.0395 −0.755263 −0.377631 0.925956i \(-0.623261\pi\)
−0.377631 + 0.925956i \(0.623261\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −36.5873 −1.61695
\(513\) −6.81544 −0.300909
\(514\) 63.7635 2.81249
\(515\) 0 0
\(516\) −13.8814 −0.611096
\(517\) 15.1183 0.664900
\(518\) 0 0
\(519\) 11.1637 0.490031
\(520\) 0 0
\(521\) −2.61265 −0.114462 −0.0572312 0.998361i \(-0.518227\pi\)
−0.0572312 + 0.998361i \(0.518227\pi\)
\(522\) 9.74944 0.426721
\(523\) −28.0032 −1.22449 −0.612247 0.790666i \(-0.709734\pi\)
−0.612247 + 0.790666i \(0.709734\pi\)
\(524\) −54.2357 −2.36930
\(525\) 0 0
\(526\) −32.6127 −1.42198
\(527\) −34.8080 −1.51626
\(528\) −16.7138 −0.727377
\(529\) −7.93440 −0.344974
\(530\) 0 0
\(531\) −0.700398 −0.0303947
\(532\) 0 0
\(533\) 11.1079 0.481137
\(534\) 0.713847 0.0308912
\(535\) 0 0
\(536\) 9.18007 0.396519
\(537\) 17.5123 0.755713
\(538\) 6.99719 0.301670
\(539\) 0 0
\(540\) 0 0
\(541\) −7.45696 −0.320600 −0.160300 0.987068i \(-0.551246\pi\)
−0.160300 + 0.987068i \(0.551246\pi\)
\(542\) −71.5168 −3.07191
\(543\) 20.2653 0.869666
\(544\) 2.44421 0.104795
\(545\) 0 0
\(546\) 0 0
\(547\) 16.7640 0.716776 0.358388 0.933573i \(-0.383326\pi\)
0.358388 + 0.933573i \(0.383326\pi\)
\(548\) −35.6443 −1.52265
\(549\) 5.00000 0.213395
\(550\) 0 0
\(551\) 27.2618 1.16139
\(552\) −18.3601 −0.781460
\(553\) 0 0
\(554\) −48.7058 −2.06931
\(555\) 0 0
\(556\) 84.0606 3.56497
\(557\) 15.0451 0.637483 0.318741 0.947842i \(-0.396740\pi\)
0.318741 + 0.947842i \(0.396740\pi\)
\(558\) 19.7629 0.836630
\(559\) −4.78674 −0.202458
\(560\) 0 0
\(561\) 19.6693 0.830438
\(562\) 17.3764 0.732980
\(563\) 21.1676 0.892108 0.446054 0.895006i \(-0.352829\pi\)
0.446054 + 0.895006i \(0.352829\pi\)
\(564\) −13.0028 −0.547517
\(565\) 0 0
\(566\) 55.7549 2.34355
\(567\) 0 0
\(568\) 56.1384 2.35552
\(569\) 44.9767 1.88552 0.942761 0.333469i \(-0.108219\pi\)
0.942761 + 0.333469i \(0.108219\pi\)
\(570\) 0 0
\(571\) 39.2238 1.64146 0.820732 0.571314i \(-0.193566\pi\)
0.820732 + 0.571314i \(0.193566\pi\)
\(572\) −24.5356 −1.02589
\(573\) 4.13592 0.172780
\(574\) 0 0
\(575\) 0 0
\(576\) −8.68344 −0.361810
\(577\) 7.06319 0.294044 0.147022 0.989133i \(-0.453031\pi\)
0.147022 + 0.989133i \(0.453031\pi\)
\(578\) 3.48249 0.144852
\(579\) −4.76640 −0.198085
\(580\) 0 0
\(581\) 0 0
\(582\) 4.89111 0.202743
\(583\) 43.3285 1.79448
\(584\) 29.1557 1.20647
\(585\) 0 0
\(586\) −8.84976 −0.365580
\(587\) −13.7629 −0.568055 −0.284028 0.958816i \(-0.591671\pi\)
−0.284028 + 0.958816i \(0.591671\pi\)
\(588\) 0 0
\(589\) 55.2618 2.27702
\(590\) 0 0
\(591\) −5.30351 −0.218157
\(592\) −42.8847 −1.76255
\(593\) −29.6309 −1.21679 −0.608397 0.793633i \(-0.708187\pi\)
−0.608397 + 0.793633i \(0.708187\pi\)
\(594\) −11.1676 −0.458212
\(595\) 0 0
\(596\) −62.2839 −2.55125
\(597\) −2.27833 −0.0932458
\(598\) −12.8556 −0.525706
\(599\) 33.1810 1.35574 0.677870 0.735181i \(-0.262903\pi\)
0.677870 + 0.735181i \(0.262903\pi\)
\(600\) 0 0
\(601\) −27.1161 −1.10609 −0.553045 0.833151i \(-0.686534\pi\)
−0.553045 + 0.833151i \(0.686534\pi\)
\(602\) 0 0
\(603\) 1.94072 0.0790323
\(604\) 19.1680 0.779935
\(605\) 0 0
\(606\) 15.3380 0.623064
\(607\) −23.5684 −0.956612 −0.478306 0.878193i \(-0.658749\pi\)
−0.478306 + 0.878193i \(0.658749\pi\)
\(608\) −3.88047 −0.157374
\(609\) 0 0
\(610\) 0 0
\(611\) −4.48376 −0.181394
\(612\) −16.9170 −0.683831
\(613\) −18.5190 −0.747977 −0.373989 0.927433i \(-0.622010\pi\)
−0.373989 + 0.927433i \(0.622010\pi\)
\(614\) −15.1771 −0.612499
\(615\) 0 0
\(616\) 0 0
\(617\) 24.5653 0.988961 0.494480 0.869189i \(-0.335358\pi\)
0.494480 + 0.869189i \(0.335358\pi\)
\(618\) 8.31489 0.334474
\(619\) −21.2720 −0.854994 −0.427497 0.904017i \(-0.640605\pi\)
−0.427497 + 0.904017i \(0.640605\pi\)
\(620\) 0 0
\(621\) −3.88144 −0.155757
\(622\) 52.7222 2.11397
\(623\) 0 0
\(624\) 4.95698 0.198438
\(625\) 0 0
\(626\) 58.2435 2.32788
\(627\) −31.2273 −1.24710
\(628\) −25.9261 −1.03457
\(629\) 50.4678 2.01228
\(630\) 0 0
\(631\) 14.3899 0.572851 0.286426 0.958102i \(-0.407533\pi\)
0.286426 + 0.958102i \(0.407533\pi\)
\(632\) −29.1742 −1.16049
\(633\) 14.8052 0.588454
\(634\) −20.8938 −0.829798
\(635\) 0 0
\(636\) −37.2657 −1.47768
\(637\) 0 0
\(638\) 44.6704 1.76852
\(639\) 11.8680 0.469491
\(640\) 0 0
\(641\) −27.7155 −1.09470 −0.547348 0.836905i \(-0.684363\pi\)
−0.547348 + 0.836905i \(0.684363\pi\)
\(642\) 7.72057 0.304707
\(643\) 44.6065 1.75911 0.879554 0.475799i \(-0.157841\pi\)
0.879554 + 0.475799i \(0.157841\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −71.3119 −2.80573
\(647\) −13.2023 −0.519037 −0.259518 0.965738i \(-0.583564\pi\)
−0.259518 + 0.965738i \(0.583564\pi\)
\(648\) 4.73024 0.185821
\(649\) −3.20911 −0.125969
\(650\) 0 0
\(651\) 0 0
\(652\) −45.1736 −1.76914
\(653\) −12.4114 −0.485697 −0.242848 0.970064i \(-0.578082\pi\)
−0.242848 + 0.970064i \(0.578082\pi\)
\(654\) −24.9430 −0.975347
\(655\) 0 0
\(656\) −29.8187 −1.16422
\(657\) 6.16368 0.240468
\(658\) 0 0
\(659\) 18.3391 0.714390 0.357195 0.934030i \(-0.383733\pi\)
0.357195 + 0.934030i \(0.383733\pi\)
\(660\) 0 0
\(661\) 42.9277 1.66969 0.834846 0.550483i \(-0.185556\pi\)
0.834846 + 0.550483i \(0.185556\pi\)
\(662\) 32.9275 1.27977
\(663\) −5.83350 −0.226555
\(664\) 61.7250 2.39540
\(665\) 0 0
\(666\) −28.6540 −1.11032
\(667\) 15.5258 0.601160
\(668\) 82.4282 3.18924
\(669\) −6.63479 −0.256516
\(670\) 0 0
\(671\) 22.9092 0.884400
\(672\) 0 0
\(673\) −17.9400 −0.691537 −0.345768 0.938320i \(-0.612382\pi\)
−0.345768 + 0.938320i \(0.612382\pi\)
\(674\) −27.0586 −1.04226
\(675\) 0 0
\(676\) −43.9526 −1.69049
\(677\) −25.7077 −0.988027 −0.494013 0.869454i \(-0.664471\pi\)
−0.494013 + 0.869454i \(0.664471\pi\)
\(678\) −7.71104 −0.296141
\(679\) 0 0
\(680\) 0 0
\(681\) 9.03168 0.346095
\(682\) 90.5504 3.46735
\(683\) 16.3100 0.624085 0.312043 0.950068i \(-0.398987\pi\)
0.312043 + 0.950068i \(0.398987\pi\)
\(684\) 26.8578 1.02693
\(685\) 0 0
\(686\) 0 0
\(687\) 16.8328 0.642211
\(688\) 12.8498 0.489893
\(689\) −12.8503 −0.489559
\(690\) 0 0
\(691\) −16.8890 −0.642489 −0.321245 0.946996i \(-0.604101\pi\)
−0.321245 + 0.946996i \(0.604101\pi\)
\(692\) −43.9930 −1.67236
\(693\) 0 0
\(694\) −66.7452 −2.53361
\(695\) 0 0
\(696\) −18.9209 −0.717197
\(697\) 35.0914 1.32918
\(698\) −15.0586 −0.569976
\(699\) 18.2929 0.691900
\(700\) 0 0
\(701\) 31.2329 1.17965 0.589825 0.807531i \(-0.299197\pi\)
0.589825 + 0.807531i \(0.299197\pi\)
\(702\) 3.31208 0.125006
\(703\) −80.1234 −3.02191
\(704\) −39.7861 −1.49950
\(705\) 0 0
\(706\) −19.0241 −0.715981
\(707\) 0 0
\(708\) 2.76008 0.103730
\(709\) 16.9064 0.634933 0.317467 0.948269i \(-0.397168\pi\)
0.317467 + 0.948269i \(0.397168\pi\)
\(710\) 0 0
\(711\) −6.16760 −0.231303
\(712\) −1.38538 −0.0519193
\(713\) 31.4720 1.17863
\(714\) 0 0
\(715\) 0 0
\(716\) −69.0112 −2.57907
\(717\) −11.6309 −0.434363
\(718\) 10.0289 0.374274
\(719\) 12.7638 0.476008 0.238004 0.971264i \(-0.423507\pi\)
0.238004 + 0.971264i \(0.423507\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 66.9061 2.48999
\(723\) −1.76961 −0.0658126
\(724\) −79.8598 −2.96797
\(725\) 0 0
\(726\) −24.3572 −0.903981
\(727\) 12.5649 0.466006 0.233003 0.972476i \(-0.425145\pi\)
0.233003 + 0.972476i \(0.425145\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −15.1219 −0.559305
\(732\) −19.7036 −0.728266
\(733\) 38.5202 1.42277 0.711387 0.702800i \(-0.248067\pi\)
0.711387 + 0.702800i \(0.248067\pi\)
\(734\) −53.1625 −1.96227
\(735\) 0 0
\(736\) −2.20996 −0.0814601
\(737\) 8.89208 0.327544
\(738\) −19.9238 −0.733404
\(739\) −4.56729 −0.168011 −0.0840053 0.996465i \(-0.526771\pi\)
−0.0840053 + 0.996465i \(0.526771\pi\)
\(740\) 0 0
\(741\) 9.26137 0.340225
\(742\) 0 0
\(743\) 36.7043 1.34655 0.673275 0.739392i \(-0.264887\pi\)
0.673275 + 0.739392i \(0.264887\pi\)
\(744\) −38.3543 −1.40614
\(745\) 0 0
\(746\) −7.16074 −0.262173
\(747\) 13.0490 0.477439
\(748\) −77.5112 −2.83409
\(749\) 0 0
\(750\) 0 0
\(751\) −54.4092 −1.98542 −0.992710 0.120530i \(-0.961541\pi\)
−0.992710 + 0.120530i \(0.961541\pi\)
\(752\) 12.0364 0.438924
\(753\) −10.7004 −0.389944
\(754\) −13.2483 −0.482475
\(755\) 0 0
\(756\) 0 0
\(757\) −13.1704 −0.478687 −0.239343 0.970935i \(-0.576932\pi\)
−0.239343 + 0.970935i \(0.576932\pi\)
\(758\) −21.5800 −0.783821
\(759\) −17.7842 −0.645524
\(760\) 0 0
\(761\) −48.2724 −1.74987 −0.874937 0.484238i \(-0.839097\pi\)
−0.874937 + 0.484238i \(0.839097\pi\)
\(762\) 0.0259259 0.000939196 0
\(763\) 0 0
\(764\) −16.2985 −0.589659
\(765\) 0 0
\(766\) −42.9919 −1.55336
\(767\) 0.951757 0.0343660
\(768\) 31.4435 1.13462
\(769\) −19.7846 −0.713452 −0.356726 0.934209i \(-0.616107\pi\)
−0.356726 + 0.934209i \(0.616107\pi\)
\(770\) 0 0
\(771\) −26.1609 −0.942161
\(772\) 18.7831 0.676017
\(773\) 15.7707 0.567233 0.283617 0.958938i \(-0.408466\pi\)
0.283617 + 0.958938i \(0.408466\pi\)
\(774\) 8.58575 0.308609
\(775\) 0 0
\(776\) −9.49228 −0.340753
\(777\) 0 0
\(778\) 44.3301 1.58931
\(779\) −55.7116 −1.99608
\(780\) 0 0
\(781\) 54.3773 1.94577
\(782\) −40.6127 −1.45231
\(783\) −4.00000 −0.142948
\(784\) 0 0
\(785\) 0 0
\(786\) 33.5451 1.19651
\(787\) −14.6937 −0.523773 −0.261886 0.965099i \(-0.584345\pi\)
−0.261886 + 0.965099i \(0.584345\pi\)
\(788\) 20.8997 0.744520
\(789\) 13.3803 0.476352
\(790\) 0 0
\(791\) 0 0
\(792\) 21.6732 0.770124
\(793\) −6.79440 −0.241276
\(794\) −25.6500 −0.910283
\(795\) 0 0
\(796\) 8.97826 0.318226
\(797\) −25.3938 −0.899493 −0.449747 0.893156i \(-0.648486\pi\)
−0.449747 + 0.893156i \(0.648486\pi\)
\(798\) 0 0
\(799\) −14.1648 −0.501114
\(800\) 0 0
\(801\) −0.292877 −0.0103483
\(802\) 8.74663 0.308854
\(803\) 28.2410 0.996604
\(804\) −7.64784 −0.269719
\(805\) 0 0
\(806\) −26.8554 −0.945941
\(807\) −2.87081 −0.101057
\(808\) −29.7668 −1.04719
\(809\) 33.3938 1.17406 0.587031 0.809564i \(-0.300297\pi\)
0.587031 + 0.809564i \(0.300297\pi\)
\(810\) 0 0
\(811\) 33.3276 1.17029 0.585145 0.810929i \(-0.301038\pi\)
0.585145 + 0.810929i \(0.301038\pi\)
\(812\) 0 0
\(813\) 29.3419 1.02907
\(814\) −131.288 −4.60164
\(815\) 0 0
\(816\) 15.6597 0.548201
\(817\) 24.0078 0.839927
\(818\) 54.9147 1.92005
\(819\) 0 0
\(820\) 0 0
\(821\) 52.1538 1.82018 0.910091 0.414409i \(-0.136012\pi\)
0.910091 + 0.414409i \(0.136012\pi\)
\(822\) 22.0462 0.768951
\(823\) 17.0345 0.593785 0.296893 0.954911i \(-0.404050\pi\)
0.296893 + 0.954911i \(0.404050\pi\)
\(824\) −16.1369 −0.562155
\(825\) 0 0
\(826\) 0 0
\(827\) −11.2054 −0.389651 −0.194826 0.980838i \(-0.562414\pi\)
−0.194826 + 0.980838i \(0.562414\pi\)
\(828\) 15.2957 0.531562
\(829\) −20.6693 −0.717874 −0.358937 0.933362i \(-0.616861\pi\)
−0.358937 + 0.933362i \(0.616861\pi\)
\(830\) 0 0
\(831\) 19.9830 0.693204
\(832\) 11.7997 0.409083
\(833\) 0 0
\(834\) −51.9920 −1.80034
\(835\) 0 0
\(836\) 123.058 4.25605
\(837\) −8.10832 −0.280264
\(838\) −96.0454 −3.31783
\(839\) 10.3486 0.357275 0.178637 0.983915i \(-0.442831\pi\)
0.178637 + 0.983915i \(0.442831\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −20.8911 −0.719955
\(843\) −7.12919 −0.245542
\(844\) −58.3432 −2.00826
\(845\) 0 0
\(846\) 8.04232 0.276501
\(847\) 0 0
\(848\) 34.4961 1.18460
\(849\) −22.8751 −0.785072
\(850\) 0 0
\(851\) −45.6309 −1.56421
\(852\) −46.7685 −1.60226
\(853\) 50.5890 1.73213 0.866067 0.499929i \(-0.166640\pi\)
0.866067 + 0.499929i \(0.166640\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −14.9835 −0.512125
\(857\) 35.8814 1.22569 0.612843 0.790204i \(-0.290026\pi\)
0.612843 + 0.790204i \(0.290026\pi\)
\(858\) 15.1754 0.518080
\(859\) −43.1850 −1.47345 −0.736726 0.676192i \(-0.763629\pi\)
−0.736726 + 0.676192i \(0.763629\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 3.12528 0.106447
\(863\) 23.6135 0.803814 0.401907 0.915681i \(-0.368348\pi\)
0.401907 + 0.915681i \(0.368348\pi\)
\(864\) 0.569365 0.0193702
\(865\) 0 0
\(866\) 12.7043 0.431710
\(867\) −1.42879 −0.0485244
\(868\) 0 0
\(869\) −28.2590 −0.958619
\(870\) 0 0
\(871\) −2.63721 −0.0893584
\(872\) 48.4073 1.63928
\(873\) −2.00672 −0.0679173
\(874\) 64.4773 2.18098
\(875\) 0 0
\(876\) −24.2894 −0.820662
\(877\) 32.8397 1.10892 0.554459 0.832211i \(-0.312925\pi\)
0.554459 + 0.832211i \(0.312925\pi\)
\(878\) −47.8407 −1.61454
\(879\) 3.63088 0.122467
\(880\) 0 0
\(881\) 9.72896 0.327777 0.163889 0.986479i \(-0.447596\pi\)
0.163889 + 0.986479i \(0.447596\pi\)
\(882\) 0 0
\(883\) 54.5366 1.83530 0.917651 0.397387i \(-0.130083\pi\)
0.917651 + 0.397387i \(0.130083\pi\)
\(884\) 22.9882 0.773177
\(885\) 0 0
\(886\) 43.9571 1.47677
\(887\) 49.0168 1.64582 0.822912 0.568169i \(-0.192348\pi\)
0.822912 + 0.568169i \(0.192348\pi\)
\(888\) 55.6094 1.86613
\(889\) 0 0
\(890\) 0 0
\(891\) 4.58184 0.153497
\(892\) 26.1459 0.875429
\(893\) 22.4882 0.752540
\(894\) 38.5230 1.28840
\(895\) 0 0
\(896\) 0 0
\(897\) 5.27442 0.176108
\(898\) −17.0547 −0.569121
\(899\) 32.4333 1.08171
\(900\) 0 0
\(901\) −40.5959 −1.35244
\(902\) −91.2875 −3.03954
\(903\) 0 0
\(904\) 14.9650 0.497728
\(905\) 0 0
\(906\) −11.8555 −0.393873
\(907\) 3.23768 0.107505 0.0537527 0.998554i \(-0.482882\pi\)
0.0537527 + 0.998554i \(0.482882\pi\)
\(908\) −35.5913 −1.18114
\(909\) −6.29288 −0.208722
\(910\) 0 0
\(911\) 17.8758 0.592252 0.296126 0.955149i \(-0.404305\pi\)
0.296126 + 0.955149i \(0.404305\pi\)
\(912\) −24.8617 −0.823252
\(913\) 59.7886 1.97872
\(914\) −92.2316 −3.05075
\(915\) 0 0
\(916\) −66.3334 −2.19172
\(917\) 0 0
\(918\) 10.4633 0.345340
\(919\) −6.26160 −0.206551 −0.103276 0.994653i \(-0.532932\pi\)
−0.103276 + 0.994653i \(0.532932\pi\)
\(920\) 0 0
\(921\) 6.22687 0.205182
\(922\) −58.7166 −1.93373
\(923\) −16.1272 −0.530833
\(924\) 0 0
\(925\) 0 0
\(926\) −56.3406 −1.85147
\(927\) −3.41143 −0.112046
\(928\) −2.27746 −0.0747613
\(929\) 9.57512 0.314149 0.157075 0.987587i \(-0.449794\pi\)
0.157075 + 0.987587i \(0.449794\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −72.0871 −2.36129
\(933\) −21.6309 −0.708163
\(934\) 30.6242 1.00205
\(935\) 0 0
\(936\) −6.42782 −0.210100
\(937\) −55.8679 −1.82512 −0.912562 0.408939i \(-0.865899\pi\)
−0.912562 + 0.408939i \(0.865899\pi\)
\(938\) 0 0
\(939\) −23.8962 −0.779822
\(940\) 0 0
\(941\) −27.2830 −0.889402 −0.444701 0.895679i \(-0.646690\pi\)
−0.444701 + 0.895679i \(0.646690\pi\)
\(942\) 16.0355 0.522464
\(943\) −31.7282 −1.03321
\(944\) −2.55494 −0.0831564
\(945\) 0 0
\(946\) 39.3386 1.27901
\(947\) 52.1671 1.69520 0.847601 0.530634i \(-0.178046\pi\)
0.847601 + 0.530634i \(0.178046\pi\)
\(948\) 24.3048 0.789383
\(949\) −8.37571 −0.271887
\(950\) 0 0
\(951\) 8.57231 0.277976
\(952\) 0 0
\(953\) −27.4644 −0.889659 −0.444829 0.895615i \(-0.646736\pi\)
−0.444829 + 0.895615i \(0.646736\pi\)
\(954\) 23.0490 0.746240
\(955\) 0 0
\(956\) 45.8341 1.48238
\(957\) −18.3274 −0.592440
\(958\) 4.94981 0.159921
\(959\) 0 0
\(960\) 0 0
\(961\) 34.7448 1.12080
\(962\) 38.9373 1.25539
\(963\) −3.16760 −0.102074
\(964\) 6.97355 0.224603
\(965\) 0 0
\(966\) 0 0
\(967\) −45.8901 −1.47573 −0.737864 0.674950i \(-0.764165\pi\)
−0.737864 + 0.674950i \(0.764165\pi\)
\(968\) 47.2706 1.51933
\(969\) 29.2579 0.939898
\(970\) 0 0
\(971\) −39.0569 −1.25339 −0.626697 0.779263i \(-0.715594\pi\)
−0.626697 + 0.779263i \(0.715594\pi\)
\(972\) −3.94072 −0.126399
\(973\) 0 0
\(974\) −67.0824 −2.14946
\(975\) 0 0
\(976\) 18.2392 0.583823
\(977\) 9.12137 0.291818 0.145909 0.989298i \(-0.453389\pi\)
0.145909 + 0.989298i \(0.453389\pi\)
\(978\) 27.9401 0.893427
\(979\) −1.34192 −0.0428879
\(980\) 0 0
\(981\) 10.2336 0.326734
\(982\) 1.11966 0.0357297
\(983\) 45.7038 1.45772 0.728862 0.684661i \(-0.240050\pi\)
0.728862 + 0.684661i \(0.240050\pi\)
\(984\) 38.6665 1.23264
\(985\) 0 0
\(986\) −41.8531 −1.33288
\(987\) 0 0
\(988\) −36.4965 −1.16111
\(989\) 13.6726 0.434764
\(990\) 0 0
\(991\) −12.7798 −0.405965 −0.202983 0.979182i \(-0.565064\pi\)
−0.202983 + 0.979182i \(0.565064\pi\)
\(992\) −4.61659 −0.146577
\(993\) −13.5095 −0.428711
\(994\) 0 0
\(995\) 0 0
\(996\) −51.4226 −1.62939
\(997\) −4.09136 −0.129575 −0.0647873 0.997899i \(-0.520637\pi\)
−0.0647873 + 0.997899i \(0.520637\pi\)
\(998\) −28.3667 −0.897934
\(999\) 11.7562 0.371949
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 3675.2.a.bq.1.4 4
5.4 even 2 3675.2.a.bx.1.1 4
7.3 odd 6 525.2.i.j.226.1 yes 8
7.5 odd 6 525.2.i.j.151.1 yes 8
7.6 odd 2 3675.2.a.br.1.4 4
35.3 even 12 525.2.r.h.499.7 16
35.12 even 12 525.2.r.h.424.7 16
35.17 even 12 525.2.r.h.499.2 16
35.19 odd 6 525.2.i.i.151.4 8
35.24 odd 6 525.2.i.i.226.4 yes 8
35.33 even 12 525.2.r.h.424.2 16
35.34 odd 2 3675.2.a.bw.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
525.2.i.i.151.4 8 35.19 odd 6
525.2.i.i.226.4 yes 8 35.24 odd 6
525.2.i.j.151.1 yes 8 7.5 odd 6
525.2.i.j.226.1 yes 8 7.3 odd 6
525.2.r.h.424.2 16 35.33 even 12
525.2.r.h.424.7 16 35.12 even 12
525.2.r.h.499.2 16 35.17 even 12
525.2.r.h.499.7 16 35.3 even 12
3675.2.a.bq.1.4 4 1.1 even 1 trivial
3675.2.a.br.1.4 4 7.6 odd 2
3675.2.a.bw.1.1 4 35.34 odd 2
3675.2.a.bx.1.1 4 5.4 even 2