Properties

Label 425.4.b.c.324.1
Level $425$
Weight $4$
Character 425.324
Analytic conductor $25.076$
Analytic rank $1$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [425,4,Mod(324,425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(425, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("425.324");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.b (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: \(25.0758117524\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 324.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 425.324
Dual form 425.4.b.c.324.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000i q^{2} +8.00000i q^{3} -1.00000 q^{4} +24.0000 q^{6} -28.0000i q^{7} -21.0000i q^{8} -37.0000 q^{9} +O(q^{10})\) \(q-3.00000i q^{2} +8.00000i q^{3} -1.00000 q^{4} +24.0000 q^{6} -28.0000i q^{7} -21.0000i q^{8} -37.0000 q^{9} -24.0000 q^{11} -8.00000i q^{12} +58.0000i q^{13} -84.0000 q^{14} -71.0000 q^{16} +17.0000i q^{17} +111.000i q^{18} -116.000 q^{19} +224.000 q^{21} +72.0000i q^{22} +60.0000i q^{23} +168.000 q^{24} +174.000 q^{26} -80.0000i q^{27} +28.0000i q^{28} -30.0000 q^{29} -172.000 q^{31} +45.0000i q^{32} -192.000i q^{33} +51.0000 q^{34} +37.0000 q^{36} -58.0000i q^{37} +348.000i q^{38} -464.000 q^{39} -342.000 q^{41} -672.000i q^{42} +148.000i q^{43} +24.0000 q^{44} +180.000 q^{46} +288.000i q^{47} -568.000i q^{48} -441.000 q^{49} -136.000 q^{51} -58.0000i q^{52} -318.000i q^{53} -240.000 q^{54} -588.000 q^{56} -928.000i q^{57} +90.0000i q^{58} -252.000 q^{59} +110.000 q^{61} +516.000i q^{62} +1036.00i q^{63} -433.000 q^{64} -576.000 q^{66} -484.000i q^{67} -17.0000i q^{68} -480.000 q^{69} -708.000 q^{71} +777.000i q^{72} -362.000i q^{73} -174.000 q^{74} +116.000 q^{76} +672.000i q^{77} +1392.00i q^{78} +484.000 q^{79} -359.000 q^{81} +1026.00i q^{82} -756.000i q^{83} -224.000 q^{84} +444.000 q^{86} -240.000i q^{87} +504.000i q^{88} +774.000 q^{89} +1624.00 q^{91} -60.0000i q^{92} -1376.00i q^{93} +864.000 q^{94} -360.000 q^{96} -382.000i q^{97} +1323.00i q^{98} +888.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 48 q^{6} - 74 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 48 q^{6} - 74 q^{9} - 48 q^{11} - 168 q^{14} - 142 q^{16} - 232 q^{19} + 448 q^{21} + 336 q^{24} + 348 q^{26} - 60 q^{29} - 344 q^{31} + 102 q^{34} + 74 q^{36} - 928 q^{39} - 684 q^{41} + 48 q^{44} + 360 q^{46} - 882 q^{49} - 272 q^{51} - 480 q^{54} - 1176 q^{56} - 504 q^{59} + 220 q^{61} - 866 q^{64} - 1152 q^{66} - 960 q^{69} - 1416 q^{71} - 348 q^{74} + 232 q^{76} + 968 q^{79} - 718 q^{81} - 448 q^{84} + 888 q^{86} + 1548 q^{89} + 3248 q^{91} + 1728 q^{94} - 720 q^{96} + 1776 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.00000i − 1.06066i −0.847791 0.530330i \(-0.822068\pi\)
0.847791 0.530330i \(-0.177932\pi\)
\(3\) 8.00000i 1.53960i 0.638285 + 0.769800i \(0.279644\pi\)
−0.638285 + 0.769800i \(0.720356\pi\)
\(4\) −1.00000 −0.125000
\(5\) 0 0
\(6\) 24.0000 1.63299
\(7\) − 28.0000i − 1.51186i −0.654654 0.755929i \(-0.727186\pi\)
0.654654 0.755929i \(-0.272814\pi\)
\(8\) − 21.0000i − 0.928078i
\(9\) −37.0000 −1.37037
\(10\) 0 0
\(11\) −24.0000 −0.657843 −0.328921 0.944357i \(-0.606685\pi\)
−0.328921 + 0.944357i \(0.606685\pi\)
\(12\) − 8.00000i − 0.192450i
\(13\) 58.0000i 1.23741i 0.785624 + 0.618704i \(0.212342\pi\)
−0.785624 + 0.618704i \(0.787658\pi\)
\(14\) −84.0000 −1.60357
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) 17.0000i 0.242536i
\(18\) 111.000i 1.45350i
\(19\) −116.000 −1.40064 −0.700322 0.713827i \(-0.746960\pi\)
−0.700322 + 0.713827i \(0.746960\pi\)
\(20\) 0 0
\(21\) 224.000 2.32766
\(22\) 72.0000i 0.697748i
\(23\) 60.0000i 0.543951i 0.962304 + 0.271975i \(0.0876769\pi\)
−0.962304 + 0.271975i \(0.912323\pi\)
\(24\) 168.000 1.42887
\(25\) 0 0
\(26\) 174.000 1.31247
\(27\) − 80.0000i − 0.570222i
\(28\) 28.0000i 0.188982i
\(29\) −30.0000 −0.192099 −0.0960493 0.995377i \(-0.530621\pi\)
−0.0960493 + 0.995377i \(0.530621\pi\)
\(30\) 0 0
\(31\) −172.000 −0.996520 −0.498260 0.867028i \(-0.666027\pi\)
−0.498260 + 0.867028i \(0.666027\pi\)
\(32\) 45.0000i 0.248592i
\(33\) − 192.000i − 1.01282i
\(34\) 51.0000 0.257248
\(35\) 0 0
\(36\) 37.0000 0.171296
\(37\) − 58.0000i − 0.257707i −0.991664 0.128853i \(-0.958870\pi\)
0.991664 0.128853i \(-0.0411296\pi\)
\(38\) 348.000i 1.48561i
\(39\) −464.000 −1.90511
\(40\) 0 0
\(41\) −342.000 −1.30272 −0.651359 0.758770i \(-0.725801\pi\)
−0.651359 + 0.758770i \(0.725801\pi\)
\(42\) − 672.000i − 2.46885i
\(43\) 148.000i 0.524879i 0.964948 + 0.262439i \(0.0845270\pi\)
−0.964948 + 0.262439i \(0.915473\pi\)
\(44\) 24.0000 0.0822304
\(45\) 0 0
\(46\) 180.000 0.576947
\(47\) 288.000i 0.893811i 0.894581 + 0.446906i \(0.147474\pi\)
−0.894581 + 0.446906i \(0.852526\pi\)
\(48\) − 568.000i − 1.70799i
\(49\) −441.000 −1.28571
\(50\) 0 0
\(51\) −136.000 −0.373408
\(52\) − 58.0000i − 0.154676i
\(53\) − 318.000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) −240.000 −0.604812
\(55\) 0 0
\(56\) −588.000 −1.40312
\(57\) − 928.000i − 2.15643i
\(58\) 90.0000i 0.203751i
\(59\) −252.000 −0.556061 −0.278031 0.960572i \(-0.589682\pi\)
−0.278031 + 0.960572i \(0.589682\pi\)
\(60\) 0 0
\(61\) 110.000 0.230886 0.115443 0.993314i \(-0.463171\pi\)
0.115443 + 0.993314i \(0.463171\pi\)
\(62\) 516.000i 1.05697i
\(63\) 1036.00i 2.07181i
\(64\) −433.000 −0.845703
\(65\) 0 0
\(66\) −576.000 −1.07425
\(67\) − 484.000i − 0.882537i −0.897375 0.441269i \(-0.854529\pi\)
0.897375 0.441269i \(-0.145471\pi\)
\(68\) − 17.0000i − 0.0303170i
\(69\) −480.000 −0.837467
\(70\) 0 0
\(71\) −708.000 −1.18344 −0.591719 0.806144i \(-0.701551\pi\)
−0.591719 + 0.806144i \(0.701551\pi\)
\(72\) 777.000i 1.27181i
\(73\) − 362.000i − 0.580396i −0.956967 0.290198i \(-0.906279\pi\)
0.956967 0.290198i \(-0.0937211\pi\)
\(74\) −174.000 −0.273339
\(75\) 0 0
\(76\) 116.000 0.175080
\(77\) 672.000i 0.994565i
\(78\) 1392.00i 2.02068i
\(79\) 484.000 0.689294 0.344647 0.938732i \(-0.387999\pi\)
0.344647 + 0.938732i \(0.387999\pi\)
\(80\) 0 0
\(81\) −359.000 −0.492455
\(82\) 1026.00i 1.38174i
\(83\) − 756.000i − 0.999780i −0.866089 0.499890i \(-0.833374\pi\)
0.866089 0.499890i \(-0.166626\pi\)
\(84\) −224.000 −0.290957
\(85\) 0 0
\(86\) 444.000 0.556718
\(87\) − 240.000i − 0.295755i
\(88\) 504.000i 0.610529i
\(89\) 774.000 0.921841 0.460920 0.887441i \(-0.347519\pi\)
0.460920 + 0.887441i \(0.347519\pi\)
\(90\) 0 0
\(91\) 1624.00 1.87079
\(92\) − 60.0000i − 0.0679938i
\(93\) − 1376.00i − 1.53424i
\(94\) 864.000 0.948030
\(95\) 0 0
\(96\) −360.000 −0.382733
\(97\) − 382.000i − 0.399858i −0.979810 0.199929i \(-0.935929\pi\)
0.979810 0.199929i \(-0.0640711\pi\)
\(98\) 1323.00i 1.36371i
\(99\) 888.000 0.901488
\(100\) 0 0
\(101\) −210.000 −0.206889 −0.103444 0.994635i \(-0.532986\pi\)
−0.103444 + 0.994635i \(0.532986\pi\)
\(102\) 408.000i 0.396059i
\(103\) 232.000i 0.221938i 0.993824 + 0.110969i \(0.0353955\pi\)
−0.993824 + 0.110969i \(0.964605\pi\)
\(104\) 1218.00 1.14841
\(105\) 0 0
\(106\) −954.000 −0.874157
\(107\) 432.000i 0.390309i 0.980773 + 0.195154i \(0.0625208\pi\)
−0.980773 + 0.195154i \(0.937479\pi\)
\(108\) 80.0000i 0.0712778i
\(109\) 1186.00 1.04219 0.521093 0.853500i \(-0.325525\pi\)
0.521093 + 0.853500i \(0.325525\pi\)
\(110\) 0 0
\(111\) 464.000 0.396765
\(112\) 1988.00i 1.67722i
\(113\) 366.000i 0.304694i 0.988327 + 0.152347i \(0.0486831\pi\)
−0.988327 + 0.152347i \(0.951317\pi\)
\(114\) −2784.00 −2.28724
\(115\) 0 0
\(116\) 30.0000 0.0240123
\(117\) − 2146.00i − 1.69571i
\(118\) 756.000i 0.589792i
\(119\) 476.000 0.366679
\(120\) 0 0
\(121\) −755.000 −0.567243
\(122\) − 330.000i − 0.244892i
\(123\) − 2736.00i − 2.00567i
\(124\) 172.000 0.124565
\(125\) 0 0
\(126\) 3108.00 2.19748
\(127\) − 472.000i − 0.329789i −0.986311 0.164895i \(-0.947272\pi\)
0.986311 0.164895i \(-0.0527284\pi\)
\(128\) 1659.00i 1.14560i
\(129\) −1184.00 −0.808104
\(130\) 0 0
\(131\) 2760.00 1.84078 0.920391 0.391000i \(-0.127871\pi\)
0.920391 + 0.391000i \(0.127871\pi\)
\(132\) 192.000i 0.126602i
\(133\) 3248.00i 2.11757i
\(134\) −1452.00 −0.936072
\(135\) 0 0
\(136\) 357.000 0.225092
\(137\) 1098.00i 0.684733i 0.939566 + 0.342367i \(0.111229\pi\)
−0.939566 + 0.342367i \(0.888771\pi\)
\(138\) 1440.00i 0.888268i
\(139\) −2528.00 −1.54261 −0.771303 0.636468i \(-0.780395\pi\)
−0.771303 + 0.636468i \(0.780395\pi\)
\(140\) 0 0
\(141\) −2304.00 −1.37611
\(142\) 2124.00i 1.25523i
\(143\) − 1392.00i − 0.814020i
\(144\) 2627.00 1.52025
\(145\) 0 0
\(146\) −1086.00 −0.615603
\(147\) − 3528.00i − 1.97949i
\(148\) 58.0000i 0.0322133i
\(149\) −1614.00 −0.887410 −0.443705 0.896173i \(-0.646336\pi\)
−0.443705 + 0.896173i \(0.646336\pi\)
\(150\) 0 0
\(151\) −3328.00 −1.79357 −0.896784 0.442468i \(-0.854103\pi\)
−0.896784 + 0.442468i \(0.854103\pi\)
\(152\) 2436.00i 1.29991i
\(153\) − 629.000i − 0.332364i
\(154\) 2016.00 1.05490
\(155\) 0 0
\(156\) 464.000 0.238139
\(157\) − 2458.00i − 1.24949i −0.780829 0.624744i \(-0.785203\pi\)
0.780829 0.624744i \(-0.214797\pi\)
\(158\) − 1452.00i − 0.731107i
\(159\) 2544.00 1.26888
\(160\) 0 0
\(161\) 1680.00 0.822376
\(162\) 1077.00i 0.522328i
\(163\) − 272.000i − 0.130704i −0.997862 0.0653518i \(-0.979183\pi\)
0.997862 0.0653518i \(-0.0208170\pi\)
\(164\) 342.000 0.162840
\(165\) 0 0
\(166\) −2268.00 −1.06043
\(167\) 3516.00i 1.62920i 0.580024 + 0.814600i \(0.303043\pi\)
−0.580024 + 0.814600i \(0.696957\pi\)
\(168\) − 4704.00i − 2.16025i
\(169\) −1167.00 −0.531179
\(170\) 0 0
\(171\) 4292.00 1.91940
\(172\) − 148.000i − 0.0656099i
\(173\) 1842.00i 0.809507i 0.914426 + 0.404753i \(0.132643\pi\)
−0.914426 + 0.404753i \(0.867357\pi\)
\(174\) −720.000 −0.313696
\(175\) 0 0
\(176\) 1704.00 0.729795
\(177\) − 2016.00i − 0.856112i
\(178\) − 2322.00i − 0.977760i
\(179\) 3516.00 1.46815 0.734073 0.679070i \(-0.237617\pi\)
0.734073 + 0.679070i \(0.237617\pi\)
\(180\) 0 0
\(181\) 3398.00 1.39542 0.697711 0.716379i \(-0.254202\pi\)
0.697711 + 0.716379i \(0.254202\pi\)
\(182\) − 4872.00i − 1.98427i
\(183\) 880.000i 0.355473i
\(184\) 1260.00 0.504828
\(185\) 0 0
\(186\) −4128.00 −1.62731
\(187\) − 408.000i − 0.159550i
\(188\) − 288.000i − 0.111726i
\(189\) −2240.00 −0.862095
\(190\) 0 0
\(191\) −2640.00 −1.00012 −0.500062 0.865990i \(-0.666689\pi\)
−0.500062 + 0.865990i \(0.666689\pi\)
\(192\) − 3464.00i − 1.30205i
\(193\) − 2882.00i − 1.07488i −0.843304 0.537438i \(-0.819392\pi\)
0.843304 0.537438i \(-0.180608\pi\)
\(194\) −1146.00 −0.424113
\(195\) 0 0
\(196\) 441.000 0.160714
\(197\) − 42.0000i − 0.0151897i −0.999971 0.00759486i \(-0.997582\pi\)
0.999971 0.00759486i \(-0.00241754\pi\)
\(198\) − 2664.00i − 0.956173i
\(199\) 3220.00 1.14703 0.573517 0.819194i \(-0.305579\pi\)
0.573517 + 0.819194i \(0.305579\pi\)
\(200\) 0 0
\(201\) 3872.00 1.35876
\(202\) 630.000i 0.219439i
\(203\) 840.000i 0.290426i
\(204\) 136.000 0.0466760
\(205\) 0 0
\(206\) 696.000 0.235401
\(207\) − 2220.00i − 0.745414i
\(208\) − 4118.00i − 1.37275i
\(209\) 2784.00 0.921403
\(210\) 0 0
\(211\) −2080.00 −0.678640 −0.339320 0.940671i \(-0.610197\pi\)
−0.339320 + 0.940671i \(0.610197\pi\)
\(212\) 318.000i 0.103020i
\(213\) − 5664.00i − 1.82202i
\(214\) 1296.00 0.413985
\(215\) 0 0
\(216\) −1680.00 −0.529211
\(217\) 4816.00i 1.50660i
\(218\) − 3558.00i − 1.10540i
\(219\) 2896.00 0.893578
\(220\) 0 0
\(221\) −986.000 −0.300116
\(222\) − 1392.00i − 0.420833i
\(223\) − 4664.00i − 1.40056i −0.713869 0.700279i \(-0.753059\pi\)
0.713869 0.700279i \(-0.246941\pi\)
\(224\) 1260.00 0.375836
\(225\) 0 0
\(226\) 1098.00 0.323176
\(227\) − 1440.00i − 0.421040i −0.977590 0.210520i \(-0.932484\pi\)
0.977590 0.210520i \(-0.0675158\pi\)
\(228\) 928.000i 0.269554i
\(229\) 1186.00 0.342241 0.171120 0.985250i \(-0.445261\pi\)
0.171120 + 0.985250i \(0.445261\pi\)
\(230\) 0 0
\(231\) −5376.00 −1.53123
\(232\) 630.000i 0.178282i
\(233\) 5334.00i 1.49975i 0.661579 + 0.749875i \(0.269887\pi\)
−0.661579 + 0.749875i \(0.730113\pi\)
\(234\) −6438.00 −1.79857
\(235\) 0 0
\(236\) 252.000 0.0695076
\(237\) 3872.00i 1.06124i
\(238\) − 1428.00i − 0.388922i
\(239\) −5328.00 −1.44201 −0.721003 0.692931i \(-0.756319\pi\)
−0.721003 + 0.692931i \(0.756319\pi\)
\(240\) 0 0
\(241\) 5618.00 1.50161 0.750803 0.660526i \(-0.229667\pi\)
0.750803 + 0.660526i \(0.229667\pi\)
\(242\) 2265.00i 0.601652i
\(243\) − 5032.00i − 1.32841i
\(244\) −110.000 −0.0288608
\(245\) 0 0
\(246\) −8208.00 −2.12733
\(247\) − 6728.00i − 1.73317i
\(248\) 3612.00i 0.924848i
\(249\) 6048.00 1.53926
\(250\) 0 0
\(251\) −2028.00 −0.509985 −0.254992 0.966943i \(-0.582073\pi\)
−0.254992 + 0.966943i \(0.582073\pi\)
\(252\) − 1036.00i − 0.258976i
\(253\) − 1440.00i − 0.357834i
\(254\) −1416.00 −0.349794
\(255\) 0 0
\(256\) 1513.00 0.369385
\(257\) − 1902.00i − 0.461648i −0.972996 0.230824i \(-0.925858\pi\)
0.972996 0.230824i \(-0.0741421\pi\)
\(258\) 3552.00i 0.857123i
\(259\) −1624.00 −0.389616
\(260\) 0 0
\(261\) 1110.00 0.263246
\(262\) − 8280.00i − 1.95244i
\(263\) 5472.00i 1.28296i 0.767141 + 0.641479i \(0.221679\pi\)
−0.767141 + 0.641479i \(0.778321\pi\)
\(264\) −4032.00 −0.939971
\(265\) 0 0
\(266\) 9744.00 2.24603
\(267\) 6192.00i 1.41927i
\(268\) 484.000i 0.110317i
\(269\) 3570.00 0.809170 0.404585 0.914500i \(-0.367416\pi\)
0.404585 + 0.914500i \(0.367416\pi\)
\(270\) 0 0
\(271\) 272.000 0.0609698 0.0304849 0.999535i \(-0.490295\pi\)
0.0304849 + 0.999535i \(0.490295\pi\)
\(272\) − 1207.00i − 0.269063i
\(273\) 12992.0i 2.88026i
\(274\) 3294.00 0.726269
\(275\) 0 0
\(276\) 480.000 0.104683
\(277\) 3830.00i 0.830767i 0.909646 + 0.415383i \(0.136353\pi\)
−0.909646 + 0.415383i \(0.863647\pi\)
\(278\) 7584.00i 1.63618i
\(279\) 6364.00 1.36560
\(280\) 0 0
\(281\) 8874.00 1.88391 0.941955 0.335740i \(-0.108986\pi\)
0.941955 + 0.335740i \(0.108986\pi\)
\(282\) 6912.00i 1.45959i
\(283\) 2632.00i 0.552849i 0.961036 + 0.276424i \(0.0891495\pi\)
−0.961036 + 0.276424i \(0.910850\pi\)
\(284\) 708.000 0.147930
\(285\) 0 0
\(286\) −4176.00 −0.863399
\(287\) 9576.00i 1.96952i
\(288\) − 1665.00i − 0.340663i
\(289\) −289.000 −0.0588235
\(290\) 0 0
\(291\) 3056.00 0.615622
\(292\) 362.000i 0.0725495i
\(293\) 6402.00i 1.27648i 0.769837 + 0.638240i \(0.220337\pi\)
−0.769837 + 0.638240i \(0.779663\pi\)
\(294\) −10584.0 −2.09956
\(295\) 0 0
\(296\) −1218.00 −0.239172
\(297\) 1920.00i 0.375117i
\(298\) 4842.00i 0.941240i
\(299\) −3480.00 −0.673089
\(300\) 0 0
\(301\) 4144.00 0.793542
\(302\) 9984.00i 1.90237i
\(303\) − 1680.00i − 0.318526i
\(304\) 8236.00 1.55384
\(305\) 0 0
\(306\) −1887.00 −0.352525
\(307\) − 8980.00i − 1.66943i −0.550681 0.834716i \(-0.685632\pi\)
0.550681 0.834716i \(-0.314368\pi\)
\(308\) − 672.000i − 0.124321i
\(309\) −1856.00 −0.341696
\(310\) 0 0
\(311\) −3972.00 −0.724217 −0.362108 0.932136i \(-0.617943\pi\)
−0.362108 + 0.932136i \(0.617943\pi\)
\(312\) 9744.00i 1.76809i
\(313\) − 4730.00i − 0.854171i −0.904211 0.427085i \(-0.859540\pi\)
0.904211 0.427085i \(-0.140460\pi\)
\(314\) −7374.00 −1.32528
\(315\) 0 0
\(316\) −484.000 −0.0861618
\(317\) − 2898.00i − 0.513463i −0.966483 0.256732i \(-0.917354\pi\)
0.966483 0.256732i \(-0.0826457\pi\)
\(318\) − 7632.00i − 1.34585i
\(319\) 720.000 0.126371
\(320\) 0 0
\(321\) −3456.00 −0.600919
\(322\) − 5040.00i − 0.872262i
\(323\) − 1972.00i − 0.339706i
\(324\) 359.000 0.0615569
\(325\) 0 0
\(326\) −816.000 −0.138632
\(327\) 9488.00i 1.60455i
\(328\) 7182.00i 1.20902i
\(329\) 8064.00 1.35132
\(330\) 0 0
\(331\) −4564.00 −0.757886 −0.378943 0.925420i \(-0.623712\pi\)
−0.378943 + 0.925420i \(0.623712\pi\)
\(332\) 756.000i 0.124973i
\(333\) 2146.00i 0.353153i
\(334\) 10548.0 1.72803
\(335\) 0 0
\(336\) −15904.0 −2.58225
\(337\) 722.000i 0.116706i 0.998296 + 0.0583529i \(0.0185849\pi\)
−0.998296 + 0.0583529i \(0.981415\pi\)
\(338\) 3501.00i 0.563400i
\(339\) −2928.00 −0.469107
\(340\) 0 0
\(341\) 4128.00 0.655553
\(342\) − 12876.0i − 2.03583i
\(343\) 2744.00i 0.431959i
\(344\) 3108.00 0.487128
\(345\) 0 0
\(346\) 5526.00 0.858612
\(347\) 5544.00i 0.857687i 0.903379 + 0.428844i \(0.141079\pi\)
−0.903379 + 0.428844i \(0.858921\pi\)
\(348\) 240.000i 0.0369694i
\(349\) −11126.0 −1.70648 −0.853239 0.521519i \(-0.825365\pi\)
−0.853239 + 0.521519i \(0.825365\pi\)
\(350\) 0 0
\(351\) 4640.00 0.705598
\(352\) − 1080.00i − 0.163535i
\(353\) − 7842.00i − 1.18240i −0.806525 0.591200i \(-0.798654\pi\)
0.806525 0.591200i \(-0.201346\pi\)
\(354\) −6048.00 −0.908044
\(355\) 0 0
\(356\) −774.000 −0.115230
\(357\) 3808.00i 0.564540i
\(358\) − 10548.0i − 1.55720i
\(359\) −5040.00 −0.740950 −0.370475 0.928842i \(-0.620805\pi\)
−0.370475 + 0.928842i \(0.620805\pi\)
\(360\) 0 0
\(361\) 6597.00 0.961802
\(362\) − 10194.0i − 1.48007i
\(363\) − 6040.00i − 0.873327i
\(364\) −1624.00 −0.233848
\(365\) 0 0
\(366\) 2640.00 0.377036
\(367\) − 8404.00i − 1.19533i −0.801747 0.597664i \(-0.796096\pi\)
0.801747 0.597664i \(-0.203904\pi\)
\(368\) − 4260.00i − 0.603445i
\(369\) 12654.0 1.78521
\(370\) 0 0
\(371\) −8904.00 −1.24602
\(372\) 1376.00i 0.191780i
\(373\) 8098.00i 1.12412i 0.827095 + 0.562062i \(0.189992\pi\)
−0.827095 + 0.562062i \(0.810008\pi\)
\(374\) −1224.00 −0.169229
\(375\) 0 0
\(376\) 6048.00 0.829526
\(377\) − 1740.00i − 0.237704i
\(378\) 6720.00i 0.914390i
\(379\) −320.000 −0.0433702 −0.0216851 0.999765i \(-0.506903\pi\)
−0.0216851 + 0.999765i \(0.506903\pi\)
\(380\) 0 0
\(381\) 3776.00 0.507744
\(382\) 7920.00i 1.06079i
\(383\) 10872.0i 1.45048i 0.688497 + 0.725239i \(0.258271\pi\)
−0.688497 + 0.725239i \(0.741729\pi\)
\(384\) −13272.0 −1.76376
\(385\) 0 0
\(386\) −8646.00 −1.14008
\(387\) − 5476.00i − 0.719278i
\(388\) 382.000i 0.0499822i
\(389\) −1374.00 −0.179086 −0.0895431 0.995983i \(-0.528541\pi\)
−0.0895431 + 0.995983i \(0.528541\pi\)
\(390\) 0 0
\(391\) −1020.00 −0.131927
\(392\) 9261.00i 1.19324i
\(393\) 22080.0i 2.83407i
\(394\) −126.000 −0.0161111
\(395\) 0 0
\(396\) −888.000 −0.112686
\(397\) − 7522.00i − 0.950928i −0.879735 0.475464i \(-0.842280\pi\)
0.879735 0.475464i \(-0.157720\pi\)
\(398\) − 9660.00i − 1.21661i
\(399\) −25984.0 −3.26022
\(400\) 0 0
\(401\) 2706.00 0.336986 0.168493 0.985703i \(-0.446110\pi\)
0.168493 + 0.985703i \(0.446110\pi\)
\(402\) − 11616.0i − 1.44118i
\(403\) − 9976.00i − 1.23310i
\(404\) 210.000 0.0258611
\(405\) 0 0
\(406\) 2520.00 0.308043
\(407\) 1392.00i 0.169530i
\(408\) 2856.00i 0.346552i
\(409\) −266.000 −0.0321586 −0.0160793 0.999871i \(-0.505118\pi\)
−0.0160793 + 0.999871i \(0.505118\pi\)
\(410\) 0 0
\(411\) −8784.00 −1.05422
\(412\) − 232.000i − 0.0277423i
\(413\) 7056.00i 0.840685i
\(414\) −6660.00 −0.790631
\(415\) 0 0
\(416\) −2610.00 −0.307610
\(417\) − 20224.0i − 2.37500i
\(418\) − 8352.00i − 0.977296i
\(419\) −2688.00 −0.313407 −0.156703 0.987646i \(-0.550087\pi\)
−0.156703 + 0.987646i \(0.550087\pi\)
\(420\) 0 0
\(421\) −13810.0 −1.59871 −0.799357 0.600857i \(-0.794826\pi\)
−0.799357 + 0.600857i \(0.794826\pi\)
\(422\) 6240.00i 0.719807i
\(423\) − 10656.0i − 1.22485i
\(424\) −6678.00 −0.764888
\(425\) 0 0
\(426\) −16992.0 −1.93255
\(427\) − 3080.00i − 0.349067i
\(428\) − 432.000i − 0.0487886i
\(429\) 11136.0 1.25327
\(430\) 0 0
\(431\) 3036.00 0.339302 0.169651 0.985504i \(-0.445736\pi\)
0.169651 + 0.985504i \(0.445736\pi\)
\(432\) 5680.00i 0.632591i
\(433\) 11422.0i 1.26768i 0.773463 + 0.633841i \(0.218523\pi\)
−0.773463 + 0.633841i \(0.781477\pi\)
\(434\) 14448.0 1.59799
\(435\) 0 0
\(436\) −1186.00 −0.130273
\(437\) − 6960.00i − 0.761881i
\(438\) − 8688.00i − 0.947782i
\(439\) 52.0000 0.00565336 0.00282668 0.999996i \(-0.499100\pi\)
0.00282668 + 0.999996i \(0.499100\pi\)
\(440\) 0 0
\(441\) 16317.0 1.76190
\(442\) 2958.00i 0.318321i
\(443\) − 3108.00i − 0.333331i −0.986014 0.166665i \(-0.946700\pi\)
0.986014 0.166665i \(-0.0533000\pi\)
\(444\) −464.000 −0.0495956
\(445\) 0 0
\(446\) −13992.0 −1.48552
\(447\) − 12912.0i − 1.36626i
\(448\) 12124.0i 1.27858i
\(449\) −6114.00 −0.642622 −0.321311 0.946974i \(-0.604124\pi\)
−0.321311 + 0.946974i \(0.604124\pi\)
\(450\) 0 0
\(451\) 8208.00 0.856984
\(452\) − 366.000i − 0.0380867i
\(453\) − 26624.0i − 2.76138i
\(454\) −4320.00 −0.446581
\(455\) 0 0
\(456\) −19488.0 −2.00134
\(457\) 4106.00i 0.420286i 0.977671 + 0.210143i \(0.0673929\pi\)
−0.977671 + 0.210143i \(0.932607\pi\)
\(458\) − 3558.00i − 0.363001i
\(459\) 1360.00 0.138299
\(460\) 0 0
\(461\) 3366.00 0.340066 0.170033 0.985438i \(-0.445613\pi\)
0.170033 + 0.985438i \(0.445613\pi\)
\(462\) 16128.0i 1.62412i
\(463\) − 896.000i − 0.0899366i −0.998988 0.0449683i \(-0.985681\pi\)
0.998988 0.0449683i \(-0.0143187\pi\)
\(464\) 2130.00 0.213109
\(465\) 0 0
\(466\) 16002.0 1.59073
\(467\) − 10236.0i − 1.01427i −0.861866 0.507137i \(-0.830704\pi\)
0.861866 0.507137i \(-0.169296\pi\)
\(468\) 2146.00i 0.211963i
\(469\) −13552.0 −1.33427
\(470\) 0 0
\(471\) 19664.0 1.92371
\(472\) 5292.00i 0.516068i
\(473\) − 3552.00i − 0.345288i
\(474\) 11616.0 1.12561
\(475\) 0 0
\(476\) −476.000 −0.0458349
\(477\) 11766.0i 1.12941i
\(478\) 15984.0i 1.52948i
\(479\) −5172.00 −0.493350 −0.246675 0.969098i \(-0.579338\pi\)
−0.246675 + 0.969098i \(0.579338\pi\)
\(480\) 0 0
\(481\) 3364.00 0.318888
\(482\) − 16854.0i − 1.59269i
\(483\) 13440.0i 1.26613i
\(484\) 755.000 0.0709053
\(485\) 0 0
\(486\) −15096.0 −1.40899
\(487\) − 15052.0i − 1.40056i −0.713870 0.700278i \(-0.753059\pi\)
0.713870 0.700278i \(-0.246941\pi\)
\(488\) − 2310.00i − 0.214280i
\(489\) 2176.00 0.201231
\(490\) 0 0
\(491\) 8700.00 0.799645 0.399822 0.916593i \(-0.369072\pi\)
0.399822 + 0.916593i \(0.369072\pi\)
\(492\) 2736.00i 0.250708i
\(493\) − 510.000i − 0.0465908i
\(494\) −20184.0 −1.83830
\(495\) 0 0
\(496\) 12212.0 1.10551
\(497\) 19824.0i 1.78919i
\(498\) − 18144.0i − 1.63263i
\(499\) 1168.00 0.104783 0.0523916 0.998627i \(-0.483316\pi\)
0.0523916 + 0.998627i \(0.483316\pi\)
\(500\) 0 0
\(501\) −28128.0 −2.50832
\(502\) 6084.00i 0.540921i
\(503\) 1740.00i 0.154240i 0.997022 + 0.0771200i \(0.0245725\pi\)
−0.997022 + 0.0771200i \(0.975428\pi\)
\(504\) 21756.0 1.92280
\(505\) 0 0
\(506\) −4320.00 −0.379540
\(507\) − 9336.00i − 0.817803i
\(508\) 472.000i 0.0412236i
\(509\) 12570.0 1.09461 0.547304 0.836934i \(-0.315654\pi\)
0.547304 + 0.836934i \(0.315654\pi\)
\(510\) 0 0
\(511\) −10136.0 −0.877476
\(512\) 8733.00i 0.753804i
\(513\) 9280.00i 0.798678i
\(514\) −5706.00 −0.489651
\(515\) 0 0
\(516\) 1184.00 0.101013
\(517\) − 6912.00i − 0.587987i
\(518\) 4872.00i 0.413250i
\(519\) −14736.0 −1.24632
\(520\) 0 0
\(521\) 11658.0 0.980319 0.490160 0.871633i \(-0.336939\pi\)
0.490160 + 0.871633i \(0.336939\pi\)
\(522\) − 3330.00i − 0.279215i
\(523\) − 13700.0i − 1.14543i −0.819755 0.572714i \(-0.805890\pi\)
0.819755 0.572714i \(-0.194110\pi\)
\(524\) −2760.00 −0.230098
\(525\) 0 0
\(526\) 16416.0 1.36078
\(527\) − 2924.00i − 0.241692i
\(528\) 13632.0i 1.12359i
\(529\) 8567.00 0.704118
\(530\) 0 0
\(531\) 9324.00 0.762010
\(532\) − 3248.00i − 0.264697i
\(533\) − 19836.0i − 1.61199i
\(534\) 18576.0 1.50536
\(535\) 0 0
\(536\) −10164.0 −0.819063
\(537\) 28128.0i 2.26036i
\(538\) − 10710.0i − 0.858254i
\(539\) 10584.0 0.845798
\(540\) 0 0
\(541\) 17822.0 1.41632 0.708159 0.706053i \(-0.249526\pi\)
0.708159 + 0.706053i \(0.249526\pi\)
\(542\) − 816.000i − 0.0646683i
\(543\) 27184.0i 2.14839i
\(544\) −765.000 −0.0602925
\(545\) 0 0
\(546\) 38976.0 3.05498
\(547\) 3800.00i 0.297032i 0.988910 + 0.148516i \(0.0474496\pi\)
−0.988910 + 0.148516i \(0.952550\pi\)
\(548\) − 1098.00i − 0.0855917i
\(549\) −4070.00 −0.316400
\(550\) 0 0
\(551\) 3480.00 0.269062
\(552\) 10080.0i 0.777234i
\(553\) − 13552.0i − 1.04212i
\(554\) 11490.0 0.881161
\(555\) 0 0
\(556\) 2528.00 0.192826
\(557\) − 10074.0i − 0.766336i −0.923679 0.383168i \(-0.874833\pi\)
0.923679 0.383168i \(-0.125167\pi\)
\(558\) − 19092.0i − 1.44844i
\(559\) −8584.00 −0.649489
\(560\) 0 0
\(561\) 3264.00 0.245644
\(562\) − 26622.0i − 1.99819i
\(563\) 15948.0i 1.19383i 0.802303 + 0.596917i \(0.203608\pi\)
−0.802303 + 0.596917i \(0.796392\pi\)
\(564\) 2304.00 0.172014
\(565\) 0 0
\(566\) 7896.00 0.586385
\(567\) 10052.0i 0.744523i
\(568\) 14868.0i 1.09832i
\(569\) −21834.0 −1.60866 −0.804331 0.594181i \(-0.797476\pi\)
−0.804331 + 0.594181i \(0.797476\pi\)
\(570\) 0 0
\(571\) −21208.0 −1.55434 −0.777169 0.629292i \(-0.783345\pi\)
−0.777169 + 0.629292i \(0.783345\pi\)
\(572\) 1392.00i 0.101753i
\(573\) − 21120.0i − 1.53979i
\(574\) 28728.0 2.08900
\(575\) 0 0
\(576\) 16021.0 1.15893
\(577\) 12530.0i 0.904039i 0.892008 + 0.452020i \(0.149296\pi\)
−0.892008 + 0.452020i \(0.850704\pi\)
\(578\) 867.000i 0.0623918i
\(579\) 23056.0 1.65488
\(580\) 0 0
\(581\) −21168.0 −1.51153
\(582\) − 9168.00i − 0.652965i
\(583\) 7632.00i 0.542170i
\(584\) −7602.00 −0.538652
\(585\) 0 0
\(586\) 19206.0 1.35391
\(587\) 2220.00i 0.156097i 0.996950 + 0.0780487i \(0.0248690\pi\)
−0.996950 + 0.0780487i \(0.975131\pi\)
\(588\) 3528.00i 0.247436i
\(589\) 19952.0 1.39577
\(590\) 0 0
\(591\) 336.000 0.0233861
\(592\) 4118.00i 0.285893i
\(593\) 25038.0i 1.73387i 0.498418 + 0.866937i \(0.333915\pi\)
−0.498418 + 0.866937i \(0.666085\pi\)
\(594\) 5760.00 0.397871
\(595\) 0 0
\(596\) 1614.00 0.110926
\(597\) 25760.0i 1.76597i
\(598\) 10440.0i 0.713919i
\(599\) −5784.00 −0.394537 −0.197269 0.980349i \(-0.563207\pi\)
−0.197269 + 0.980349i \(0.563207\pi\)
\(600\) 0 0
\(601\) −4198.00 −0.284925 −0.142463 0.989800i \(-0.545502\pi\)
−0.142463 + 0.989800i \(0.545502\pi\)
\(602\) − 12432.0i − 0.841679i
\(603\) 17908.0i 1.20940i
\(604\) 3328.00 0.224196
\(605\) 0 0
\(606\) −5040.00 −0.337848
\(607\) − 12124.0i − 0.810705i −0.914160 0.405353i \(-0.867149\pi\)
0.914160 0.405353i \(-0.132851\pi\)
\(608\) − 5220.00i − 0.348189i
\(609\) −6720.00 −0.447140
\(610\) 0 0
\(611\) −16704.0 −1.10601
\(612\) 629.000i 0.0415455i
\(613\) − 7454.00i − 0.491133i −0.969380 0.245566i \(-0.921026\pi\)
0.969380 0.245566i \(-0.0789739\pi\)
\(614\) −26940.0 −1.77070
\(615\) 0 0
\(616\) 14112.0 0.923034
\(617\) 28842.0i 1.88190i 0.338539 + 0.940952i \(0.390067\pi\)
−0.338539 + 0.940952i \(0.609933\pi\)
\(618\) 5568.00i 0.362424i
\(619\) 17224.0 1.11840 0.559201 0.829032i \(-0.311108\pi\)
0.559201 + 0.829032i \(0.311108\pi\)
\(620\) 0 0
\(621\) 4800.00 0.310173
\(622\) 11916.0i 0.768148i
\(623\) − 21672.0i − 1.39369i
\(624\) 32944.0 2.11349
\(625\) 0 0
\(626\) −14190.0 −0.905985
\(627\) 22272.0i 1.41859i
\(628\) 2458.00i 0.156186i
\(629\) 986.000 0.0625030
\(630\) 0 0
\(631\) −12448.0 −0.785336 −0.392668 0.919680i \(-0.628448\pi\)
−0.392668 + 0.919680i \(0.628448\pi\)
\(632\) − 10164.0i − 0.639719i
\(633\) − 16640.0i − 1.04484i
\(634\) −8694.00 −0.544610
\(635\) 0 0
\(636\) −2544.00 −0.158610
\(637\) − 25578.0i − 1.59095i
\(638\) − 2160.00i − 0.134036i
\(639\) 26196.0 1.62175
\(640\) 0 0
\(641\) −25182.0 −1.55168 −0.775842 0.630927i \(-0.782675\pi\)
−0.775842 + 0.630927i \(0.782675\pi\)
\(642\) 10368.0i 0.637371i
\(643\) − 17048.0i − 1.04558i −0.852462 0.522790i \(-0.824891\pi\)
0.852462 0.522790i \(-0.175109\pi\)
\(644\) −1680.00 −0.102797
\(645\) 0 0
\(646\) −5916.00 −0.360313
\(647\) 7128.00i 0.433123i 0.976269 + 0.216562i \(0.0694842\pi\)
−0.976269 + 0.216562i \(0.930516\pi\)
\(648\) 7539.00i 0.457037i
\(649\) 6048.00 0.365801
\(650\) 0 0
\(651\) −38528.0 −2.31956
\(652\) 272.000i 0.0163379i
\(653\) − 18462.0i − 1.10639i −0.833051 0.553196i \(-0.813408\pi\)
0.833051 0.553196i \(-0.186592\pi\)
\(654\) 28464.0 1.70188
\(655\) 0 0
\(656\) 24282.0 1.44520
\(657\) 13394.0i 0.795357i
\(658\) − 24192.0i − 1.43329i
\(659\) −28092.0 −1.66056 −0.830280 0.557347i \(-0.811819\pi\)
−0.830280 + 0.557347i \(0.811819\pi\)
\(660\) 0 0
\(661\) 10910.0 0.641982 0.320991 0.947082i \(-0.395984\pi\)
0.320991 + 0.947082i \(0.395984\pi\)
\(662\) 13692.0i 0.803859i
\(663\) − 7888.00i − 0.462058i
\(664\) −15876.0 −0.927874
\(665\) 0 0
\(666\) 6438.00 0.374576
\(667\) − 1800.00i − 0.104492i
\(668\) − 3516.00i − 0.203650i
\(669\) 37312.0 2.15630
\(670\) 0 0
\(671\) −2640.00 −0.151887
\(672\) 10080.0i 0.578638i
\(673\) 28414.0i 1.62746i 0.581244 + 0.813729i \(0.302566\pi\)
−0.581244 + 0.813729i \(0.697434\pi\)
\(674\) 2166.00 0.123785
\(675\) 0 0
\(676\) 1167.00 0.0663974
\(677\) − 6042.00i − 0.343003i −0.985184 0.171501i \(-0.945138\pi\)
0.985184 0.171501i \(-0.0548618\pi\)
\(678\) 8784.00i 0.497563i
\(679\) −10696.0 −0.604528
\(680\) 0 0
\(681\) 11520.0 0.648234
\(682\) − 12384.0i − 0.695319i
\(683\) − 34752.0i − 1.94692i −0.228851 0.973461i \(-0.573497\pi\)
0.228851 0.973461i \(-0.426503\pi\)
\(684\) −4292.00 −0.239925
\(685\) 0 0
\(686\) 8232.00 0.458162
\(687\) 9488.00i 0.526914i
\(688\) − 10508.0i − 0.582287i
\(689\) 18444.0 1.01983
\(690\) 0 0
\(691\) 18320.0 1.00858 0.504288 0.863536i \(-0.331755\pi\)
0.504288 + 0.863536i \(0.331755\pi\)
\(692\) − 1842.00i − 0.101188i
\(693\) − 24864.0i − 1.36292i
\(694\) 16632.0 0.909715
\(695\) 0 0
\(696\) −5040.00 −0.274484
\(697\) − 5814.00i − 0.315955i
\(698\) 33378.0i 1.80999i
\(699\) −42672.0 −2.30902
\(700\) 0 0
\(701\) −22890.0 −1.23330 −0.616650 0.787237i \(-0.711511\pi\)
−0.616650 + 0.787237i \(0.711511\pi\)
\(702\) − 13920.0i − 0.748400i
\(703\) 6728.00i 0.360955i
\(704\) 10392.0 0.556340
\(705\) 0 0
\(706\) −23526.0 −1.25413
\(707\) 5880.00i 0.312787i
\(708\) 2016.00i 0.107014i
\(709\) −22886.0 −1.21227 −0.606137 0.795361i \(-0.707282\pi\)
−0.606137 + 0.795361i \(0.707282\pi\)
\(710\) 0 0
\(711\) −17908.0 −0.944589
\(712\) − 16254.0i − 0.855540i
\(713\) − 10320.0i − 0.542058i
\(714\) 11424.0 0.598785
\(715\) 0 0
\(716\) −3516.00 −0.183518
\(717\) − 42624.0i − 2.22011i
\(718\) 15120.0i 0.785896i
\(719\) 13452.0 0.697740 0.348870 0.937171i \(-0.386565\pi\)
0.348870 + 0.937171i \(0.386565\pi\)
\(720\) 0 0
\(721\) 6496.00 0.335539
\(722\) − 19791.0i − 1.02015i
\(723\) 44944.0i 2.31187i
\(724\) −3398.00 −0.174428
\(725\) 0 0
\(726\) −18120.0 −0.926303
\(727\) − 27304.0i − 1.39292i −0.717598 0.696458i \(-0.754758\pi\)
0.717598 0.696458i \(-0.245242\pi\)
\(728\) − 34104.0i − 1.73623i
\(729\) 30563.0 1.55276
\(730\) 0 0
\(731\) −2516.00 −0.127302
\(732\) − 880.000i − 0.0444341i
\(733\) − 24470.0i − 1.23304i −0.787338 0.616521i \(-0.788541\pi\)
0.787338 0.616521i \(-0.211459\pi\)
\(734\) −25212.0 −1.26784
\(735\) 0 0
\(736\) −2700.00 −0.135222
\(737\) 11616.0i 0.580571i
\(738\) − 37962.0i − 1.89350i
\(739\) −35252.0 −1.75476 −0.877379 0.479798i \(-0.840710\pi\)
−0.877379 + 0.479798i \(0.840710\pi\)
\(740\) 0 0
\(741\) 53824.0 2.66839
\(742\) 26712.0i 1.32160i
\(743\) − 1548.00i − 0.0764342i −0.999269 0.0382171i \(-0.987832\pi\)
0.999269 0.0382171i \(-0.0121678\pi\)
\(744\) −28896.0 −1.42390
\(745\) 0 0
\(746\) 24294.0 1.19231
\(747\) 27972.0i 1.37007i
\(748\) 408.000i 0.0199438i
\(749\) 12096.0 0.590091
\(750\) 0 0
\(751\) 2948.00 0.143241 0.0716205 0.997432i \(-0.477183\pi\)
0.0716205 + 0.997432i \(0.477183\pi\)
\(752\) − 20448.0i − 0.991572i
\(753\) − 16224.0i − 0.785173i
\(754\) −5220.00 −0.252124
\(755\) 0 0
\(756\) 2240.00 0.107762
\(757\) − 754.000i − 0.0362016i −0.999836 0.0181008i \(-0.994238\pi\)
0.999836 0.0181008i \(-0.00576198\pi\)
\(758\) 960.000i 0.0460010i
\(759\) 11520.0 0.550922
\(760\) 0 0
\(761\) −41574.0 −1.98036 −0.990182 0.139787i \(-0.955358\pi\)
−0.990182 + 0.139787i \(0.955358\pi\)
\(762\) − 11328.0i − 0.538543i
\(763\) − 33208.0i − 1.57564i
\(764\) 2640.00 0.125016
\(765\) 0 0
\(766\) 32616.0 1.53846
\(767\) − 14616.0i − 0.688075i
\(768\) 12104.0i 0.568705i
\(769\) 15118.0 0.708932 0.354466 0.935069i \(-0.384663\pi\)
0.354466 + 0.935069i \(0.384663\pi\)
\(770\) 0 0
\(771\) 15216.0 0.710753
\(772\) 2882.00i 0.134359i
\(773\) − 23550.0i − 1.09578i −0.836552 0.547888i \(-0.815432\pi\)
0.836552 0.547888i \(-0.184568\pi\)
\(774\) −16428.0 −0.762910
\(775\) 0 0
\(776\) −8022.00 −0.371099
\(777\) − 12992.0i − 0.599853i
\(778\) 4122.00i 0.189950i
\(779\) 39672.0 1.82464
\(780\) 0 0
\(781\) 16992.0 0.778517
\(782\) 3060.00i 0.139930i
\(783\) 2400.00i 0.109539i
\(784\) 31311.0 1.42634
\(785\) 0 0
\(786\) 66240.0 3.00598
\(787\) 5240.00i 0.237339i 0.992934 + 0.118670i \(0.0378629\pi\)
−0.992934 + 0.118670i \(0.962137\pi\)
\(788\) 42.0000i 0.00189872i
\(789\) −43776.0 −1.97524
\(790\) 0 0
\(791\) 10248.0 0.460654
\(792\) − 18648.0i − 0.836651i
\(793\) 6380.00i 0.285700i
\(794\) −22566.0 −1.00861
\(795\) 0 0
\(796\) −3220.00 −0.143379
\(797\) 5526.00i 0.245597i 0.992432 + 0.122799i \(0.0391869\pi\)
−0.992432 + 0.122799i \(0.960813\pi\)
\(798\) 77952.0i 3.45798i
\(799\) −4896.00 −0.216781
\(800\) 0 0
\(801\) −28638.0 −1.26326
\(802\) − 8118.00i − 0.357427i
\(803\) 8688.00i 0.381809i
\(804\) −3872.00 −0.169844
\(805\) 0 0
\(806\) −29928.0 −1.30790
\(807\) 28560.0i 1.24580i
\(808\) 4410.00i 0.192009i
\(809\) 438.000 0.0190349 0.00951747 0.999955i \(-0.496970\pi\)
0.00951747 + 0.999955i \(0.496970\pi\)
\(810\) 0 0
\(811\) −30448.0 −1.31834 −0.659170 0.751994i \(-0.729092\pi\)
−0.659170 + 0.751994i \(0.729092\pi\)
\(812\) − 840.000i − 0.0363032i
\(813\) 2176.00i 0.0938692i
\(814\) 4176.00 0.179814
\(815\) 0 0
\(816\) 9656.00 0.414250
\(817\) − 17168.0i − 0.735168i
\(818\) 798.000i 0.0341093i
\(819\) −60088.0 −2.56367
\(820\) 0 0
\(821\) −21930.0 −0.932232 −0.466116 0.884724i \(-0.654347\pi\)
−0.466116 + 0.884724i \(0.654347\pi\)
\(822\) 26352.0i 1.11816i
\(823\) 27436.0i 1.16204i 0.813889 + 0.581020i \(0.197346\pi\)
−0.813889 + 0.581020i \(0.802654\pi\)
\(824\) 4872.00 0.205976
\(825\) 0 0
\(826\) 21168.0 0.891681
\(827\) − 17832.0i − 0.749794i −0.927067 0.374897i \(-0.877678\pi\)
0.927067 0.374897i \(-0.122322\pi\)
\(828\) 2220.00i 0.0931767i
\(829\) 4090.00 0.171353 0.0856765 0.996323i \(-0.472695\pi\)
0.0856765 + 0.996323i \(0.472695\pi\)
\(830\) 0 0
\(831\) −30640.0 −1.27905
\(832\) − 25114.0i − 1.04648i
\(833\) − 7497.00i − 0.311832i
\(834\) −60672.0 −2.51906
\(835\) 0 0
\(836\) −2784.00 −0.115175
\(837\) 13760.0i 0.568238i
\(838\) 8064.00i 0.332418i
\(839\) 2508.00 0.103201 0.0516006 0.998668i \(-0.483568\pi\)
0.0516006 + 0.998668i \(0.483568\pi\)
\(840\) 0 0
\(841\) −23489.0 −0.963098
\(842\) 41430.0i 1.69569i
\(843\) 70992.0i 2.90047i
\(844\) 2080.00 0.0848300
\(845\) 0 0
\(846\) −31968.0 −1.29915
\(847\) 21140.0i 0.857590i
\(848\) 22578.0i 0.914306i
\(849\) −21056.0 −0.851166
\(850\) 0 0
\(851\) 3480.00 0.140180
\(852\) 5664.00i 0.227753i
\(853\) 42442.0i 1.70362i 0.523852 + 0.851809i \(0.324494\pi\)
−0.523852 + 0.851809i \(0.675506\pi\)
\(854\) −9240.00 −0.370242
\(855\) 0 0
\(856\) 9072.00 0.362237
\(857\) 32730.0i 1.30459i 0.757964 + 0.652296i \(0.226194\pi\)
−0.757964 + 0.652296i \(0.773806\pi\)
\(858\) − 33408.0i − 1.32929i
\(859\) 6148.00 0.244199 0.122100 0.992518i \(-0.461037\pi\)
0.122100 + 0.992518i \(0.461037\pi\)
\(860\) 0 0
\(861\) −76608.0 −3.03228
\(862\) − 9108.00i − 0.359884i
\(863\) 22512.0i 0.887969i 0.896034 + 0.443985i \(0.146436\pi\)
−0.896034 + 0.443985i \(0.853564\pi\)
\(864\) 3600.00 0.141753
\(865\) 0 0
\(866\) 34266.0 1.34458
\(867\) − 2312.00i − 0.0905647i
\(868\) − 4816.00i − 0.188325i
\(869\) −11616.0 −0.453447
\(870\) 0 0
\(871\) 28072.0 1.09206
\(872\) − 24906.0i − 0.967229i
\(873\) 14134.0i 0.547954i
\(874\) −20880.0 −0.808097
\(875\) 0 0
\(876\) −2896.00 −0.111697
\(877\) 9182.00i 0.353539i 0.984252 + 0.176770i \(0.0565648\pi\)
−0.984252 + 0.176770i \(0.943435\pi\)
\(878\) − 156.000i − 0.00599629i
\(879\) −51216.0 −1.96527
\(880\) 0 0
\(881\) 28530.0 1.09103 0.545517 0.838100i \(-0.316334\pi\)
0.545517 + 0.838100i \(0.316334\pi\)
\(882\) − 48951.0i − 1.86878i
\(883\) 12436.0i 0.473958i 0.971515 + 0.236979i \(0.0761572\pi\)
−0.971515 + 0.236979i \(0.923843\pi\)
\(884\) 986.000 0.0375144
\(885\) 0 0
\(886\) −9324.00 −0.353551
\(887\) 7404.00i 0.280273i 0.990132 + 0.140136i \(0.0447541\pi\)
−0.990132 + 0.140136i \(0.955246\pi\)
\(888\) − 9744.00i − 0.368229i
\(889\) −13216.0 −0.498594
\(890\) 0 0
\(891\) 8616.00 0.323958
\(892\) 4664.00i 0.175070i
\(893\) − 33408.0i − 1.25191i
\(894\) −38736.0 −1.44913
\(895\) 0 0
\(896\) 46452.0 1.73198
\(897\) − 27840.0i − 1.03629i
\(898\) 18342.0i 0.681604i
\(899\) 5160.00 0.191430
\(900\) 0 0
\(901\) 5406.00 0.199889
\(902\) − 24624.0i − 0.908968i
\(903\) 33152.0i 1.22174i
\(904\) 7686.00 0.282779
\(905\) 0 0
\(906\) −79872.0 −2.92888
\(907\) 15368.0i 0.562609i 0.959619 + 0.281304i \(0.0907670\pi\)
−0.959619 + 0.281304i \(0.909233\pi\)
\(908\) 1440.00i 0.0526300i
\(909\) 7770.00 0.283514
\(910\) 0 0
\(911\) 27276.0 0.991980 0.495990 0.868328i \(-0.334805\pi\)
0.495990 + 0.868328i \(0.334805\pi\)
\(912\) 65888.0i 2.39229i
\(913\) 18144.0i 0.657699i
\(914\) 12318.0 0.445780
\(915\) 0 0
\(916\) −1186.00 −0.0427801
\(917\) − 77280.0i − 2.78300i
\(918\) − 4080.00i − 0.146689i
\(919\) 46456.0 1.66751 0.833755 0.552134i \(-0.186186\pi\)
0.833755 + 0.552134i \(0.186186\pi\)
\(920\) 0 0
\(921\) 71840.0 2.57026
\(922\) − 10098.0i − 0.360694i
\(923\) − 41064.0i − 1.46440i
\(924\) 5376.00 0.191404
\(925\) 0 0
\(926\) −2688.00 −0.0953922
\(927\) − 8584.00i − 0.304138i
\(928\) − 1350.00i − 0.0477542i
\(929\) −13026.0 −0.460031 −0.230016 0.973187i \(-0.573878\pi\)
−0.230016 + 0.973187i \(0.573878\pi\)
\(930\) 0 0
\(931\) 51156.0 1.80083
\(932\) − 5334.00i − 0.187469i
\(933\) − 31776.0i − 1.11500i
\(934\) −30708.0 −1.07580
\(935\) 0 0
\(936\) −45066.0 −1.57375
\(937\) 26330.0i 0.917997i 0.888437 + 0.458999i \(0.151792\pi\)
−0.888437 + 0.458999i \(0.848208\pi\)
\(938\) 40656.0i 1.41521i
\(939\) 37840.0 1.31508
\(940\) 0 0
\(941\) 28254.0 0.978803 0.489402 0.872058i \(-0.337215\pi\)
0.489402 + 0.872058i \(0.337215\pi\)
\(942\) − 58992.0i − 2.04041i
\(943\) − 20520.0i − 0.708614i
\(944\) 17892.0 0.616880
\(945\) 0 0
\(946\) −10656.0 −0.366233
\(947\) 49272.0i 1.69073i 0.534186 + 0.845367i \(0.320618\pi\)
−0.534186 + 0.845367i \(0.679382\pi\)
\(948\) − 3872.00i − 0.132655i
\(949\) 20996.0 0.718187
\(950\) 0 0
\(951\) 23184.0 0.790529
\(952\) − 9996.00i − 0.340307i
\(953\) − 32922.0i − 1.11904i −0.828816 0.559522i \(-0.810985\pi\)
0.828816 0.559522i \(-0.189015\pi\)
\(954\) 35298.0 1.19792
\(955\) 0 0
\(956\) 5328.00 0.180251
\(957\) 5760.00i 0.194560i
\(958\) 15516.0i 0.523277i
\(959\) 30744.0 1.03522
\(960\) 0 0
\(961\) −207.000 −0.00694841
\(962\) − 10092.0i − 0.338232i
\(963\) − 15984.0i − 0.534867i
\(964\) −5618.00 −0.187701
\(965\) 0 0
\(966\) 40320.0 1.34293
\(967\) − 1168.00i − 0.0388421i −0.999811 0.0194211i \(-0.993818\pi\)
0.999811 0.0194211i \(-0.00618231\pi\)
\(968\) 15855.0i 0.526445i
\(969\) 15776.0 0.523011
\(970\) 0 0
\(971\) −19812.0 −0.654786 −0.327393 0.944888i \(-0.606170\pi\)
−0.327393 + 0.944888i \(0.606170\pi\)
\(972\) 5032.00i 0.166051i
\(973\) 70784.0i 2.33220i
\(974\) −45156.0 −1.48551
\(975\) 0 0
\(976\) −7810.00 −0.256139
\(977\) − 28494.0i − 0.933064i −0.884504 0.466532i \(-0.845503\pi\)
0.884504 0.466532i \(-0.154497\pi\)
\(978\) − 6528.00i − 0.213438i
\(979\) −18576.0 −0.606426
\(980\) 0 0
\(981\) −43882.0 −1.42818
\(982\) − 26100.0i − 0.848151i
\(983\) 42708.0i 1.38573i 0.721067 + 0.692866i \(0.243652\pi\)
−0.721067 + 0.692866i \(0.756348\pi\)
\(984\) −57456.0 −1.86141
\(985\) 0 0
\(986\) −1530.00 −0.0494170
\(987\) 64512.0i 2.08049i
\(988\) 6728.00i 0.216646i
\(989\) −8880.00 −0.285508
\(990\) 0 0
\(991\) −29500.0 −0.945609 −0.472804 0.881167i \(-0.656758\pi\)
−0.472804 + 0.881167i \(0.656758\pi\)
\(992\) − 7740.00i − 0.247727i
\(993\) − 36512.0i − 1.16684i
\(994\) 59472.0 1.89772
\(995\) 0 0
\(996\) −6048.00 −0.192408
\(997\) − 9322.00i − 0.296119i −0.988978 0.148060i \(-0.952697\pi\)
0.988978 0.148060i \(-0.0473027\pi\)
\(998\) − 3504.00i − 0.111139i
\(999\) −4640.00 −0.146950
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 425.4.b.c.324.1 2
5.2 odd 4 425.4.a.d.1.1 1
5.3 odd 4 17.4.a.a.1.1 1
5.4 even 2 inner 425.4.b.c.324.2 2
15.8 even 4 153.4.a.d.1.1 1
20.3 even 4 272.4.a.d.1.1 1
35.13 even 4 833.4.a.a.1.1 1
40.3 even 4 1088.4.a.a.1.1 1
40.13 odd 4 1088.4.a.l.1.1 1
55.43 even 4 2057.4.a.d.1.1 1
60.23 odd 4 2448.4.a.f.1.1 1
85.13 odd 4 289.4.b.a.288.1 2
85.33 odd 4 289.4.a.a.1.1 1
85.38 odd 4 289.4.b.a.288.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.a.a.1.1 1 5.3 odd 4
153.4.a.d.1.1 1 15.8 even 4
272.4.a.d.1.1 1 20.3 even 4
289.4.a.a.1.1 1 85.33 odd 4
289.4.b.a.288.1 2 85.13 odd 4
289.4.b.a.288.2 2 85.38 odd 4
425.4.a.d.1.1 1 5.2 odd 4
425.4.b.c.324.1 2 1.1 even 1 trivial
425.4.b.c.324.2 2 5.4 even 2 inner
833.4.a.a.1.1 1 35.13 even 4
1088.4.a.a.1.1 1 40.3 even 4
1088.4.a.l.1.1 1 40.13 odd 4
2057.4.a.d.1.1 1 55.43 even 4
2448.4.a.f.1.1 1 60.23 odd 4