Properties

Label 225.3.d.a.224.3
Level $225$
Weight $3$
Character 225.224
Analytic conductor $6.131$
Analytic rank $0$
Dimension $4$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,3,Mod(224,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.224");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 225.d (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: \(6.13080594811\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 224.3
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 225.224
Dual form 225.3.d.a.224.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421 q^{2} -2.00000 q^{4} -9.00000i q^{7} -8.48528 q^{8} -18.3848i q^{11} -1.00000i q^{13} -12.7279i q^{14} -4.00000 q^{16} +26.8701 q^{17} -25.0000 q^{19} -26.0000i q^{22} +4.24264 q^{23} -1.41421i q^{26} +18.0000i q^{28} -26.8701i q^{29} -39.0000 q^{31} +28.2843 q^{32} +38.0000 q^{34} +32.0000i q^{37} -35.3553 q^{38} +5.65685i q^{41} +23.0000i q^{43} +36.7696i q^{44} +6.00000 q^{46} -32.5269 q^{47} -32.0000 q^{49} +2.00000i q^{52} +96.1665 q^{53} +76.3675i q^{56} -38.0000i q^{58} +9.89949i q^{59} +73.0000 q^{61} -55.1543 q^{62} +56.0000 q^{64} +63.0000i q^{67} -53.7401 q^{68} -62.2254i q^{71} -136.000i q^{73} +45.2548i q^{74} +50.0000 q^{76} -165.463 q^{77} +24.0000 q^{79} +8.00000i q^{82} +46.6690 q^{83} +32.5269i q^{86} +156.000i q^{88} +101.823i q^{89} -9.00000 q^{91} -8.48528 q^{92} -46.0000 q^{94} -7.00000i q^{97} -45.2548 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} - 16 q^{16} - 100 q^{19} - 156 q^{31} + 152 q^{34} + 24 q^{46} - 128 q^{49} + 292 q^{61} + 224 q^{64} + 200 q^{76} + 96 q^{79} - 36 q^{91} - 184 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421 0.707107 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 0 0
\(4\) −2.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 9.00000i − 1.28571i −0.765986 0.642857i \(-0.777749\pi\)
0.765986 0.642857i \(-0.222251\pi\)
\(8\) −8.48528 −1.06066
\(9\) 0 0
\(10\) 0 0
\(11\) − 18.3848i − 1.67134i −0.549229 0.835672i \(-0.685079\pi\)
0.549229 0.835672i \(-0.314921\pi\)
\(12\) 0 0
\(13\) − 1.00000i − 0.0769231i −0.999260 0.0384615i \(-0.987754\pi\)
0.999260 0.0384615i \(-0.0122457\pi\)
\(14\) − 12.7279i − 0.909137i
\(15\) 0 0
\(16\) −4.00000 −0.250000
\(17\) 26.8701 1.58059 0.790296 0.612725i \(-0.209927\pi\)
0.790296 + 0.612725i \(0.209927\pi\)
\(18\) 0 0
\(19\) −25.0000 −1.31579 −0.657895 0.753110i \(-0.728553\pi\)
−0.657895 + 0.753110i \(0.728553\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 26.0000i − 1.18182i
\(23\) 4.24264 0.184463 0.0922313 0.995738i \(-0.470600\pi\)
0.0922313 + 0.995738i \(0.470600\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) − 1.41421i − 0.0543928i
\(27\) 0 0
\(28\) 18.0000i 0.642857i
\(29\) − 26.8701i − 0.926554i −0.886214 0.463277i \(-0.846674\pi\)
0.886214 0.463277i \(-0.153326\pi\)
\(30\) 0 0
\(31\) −39.0000 −1.25806 −0.629032 0.777379i \(-0.716549\pi\)
−0.629032 + 0.777379i \(0.716549\pi\)
\(32\) 28.2843 0.883883
\(33\) 0 0
\(34\) 38.0000 1.11765
\(35\) 0 0
\(36\) 0 0
\(37\) 32.0000i 0.864865i 0.901666 + 0.432432i \(0.142345\pi\)
−0.901666 + 0.432432i \(0.857655\pi\)
\(38\) −35.3553 −0.930404
\(39\) 0 0
\(40\) 0 0
\(41\) 5.65685i 0.137972i 0.997618 + 0.0689860i \(0.0219764\pi\)
−0.997618 + 0.0689860i \(0.978024\pi\)
\(42\) 0 0
\(43\) 23.0000i 0.534884i 0.963574 + 0.267442i \(0.0861783\pi\)
−0.963574 + 0.267442i \(0.913822\pi\)
\(44\) 36.7696i 0.835672i
\(45\) 0 0
\(46\) 6.00000 0.130435
\(47\) −32.5269 −0.692062 −0.346031 0.938223i \(-0.612471\pi\)
−0.346031 + 0.938223i \(0.612471\pi\)
\(48\) 0 0
\(49\) −32.0000 −0.653061
\(50\) 0 0
\(51\) 0 0
\(52\) 2.00000i 0.0384615i
\(53\) 96.1665 1.81446 0.907231 0.420632i \(-0.138192\pi\)
0.907231 + 0.420632i \(0.138192\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 76.3675i 1.36371i
\(57\) 0 0
\(58\) − 38.0000i − 0.655172i
\(59\) 9.89949i 0.167788i 0.996475 + 0.0838940i \(0.0267357\pi\)
−0.996475 + 0.0838940i \(0.973264\pi\)
\(60\) 0 0
\(61\) 73.0000 1.19672 0.598361 0.801227i \(-0.295819\pi\)
0.598361 + 0.801227i \(0.295819\pi\)
\(62\) −55.1543 −0.889586
\(63\) 0 0
\(64\) 56.0000 0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) 63.0000i 0.940299i 0.882587 + 0.470149i \(0.155800\pi\)
−0.882587 + 0.470149i \(0.844200\pi\)
\(68\) −53.7401 −0.790296
\(69\) 0 0
\(70\) 0 0
\(71\) − 62.2254i − 0.876414i −0.898874 0.438207i \(-0.855614\pi\)
0.898874 0.438207i \(-0.144386\pi\)
\(72\) 0 0
\(73\) − 136.000i − 1.86301i −0.363724 0.931507i \(-0.618495\pi\)
0.363724 0.931507i \(-0.381505\pi\)
\(74\) 45.2548i 0.611552i
\(75\) 0 0
\(76\) 50.0000 0.657895
\(77\) −165.463 −2.14887
\(78\) 0 0
\(79\) 24.0000 0.303797 0.151899 0.988396i \(-0.451461\pi\)
0.151899 + 0.988396i \(0.451461\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 8.00000i 0.0975610i
\(83\) 46.6690 0.562278 0.281139 0.959667i \(-0.409288\pi\)
0.281139 + 0.959667i \(0.409288\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 32.5269i 0.378220i
\(87\) 0 0
\(88\) 156.000i 1.77273i
\(89\) 101.823i 1.14408i 0.820225 + 0.572041i \(0.193848\pi\)
−0.820225 + 0.572041i \(0.806152\pi\)
\(90\) 0 0
\(91\) −9.00000 −0.0989011
\(92\) −8.48528 −0.0922313
\(93\) 0 0
\(94\) −46.0000 −0.489362
\(95\) 0 0
\(96\) 0 0
\(97\) − 7.00000i − 0.0721649i −0.999349 0.0360825i \(-0.988512\pi\)
0.999349 0.0360825i \(-0.0114879\pi\)
\(98\) −45.2548 −0.461784
\(99\) 0 0
\(100\) 0 0
\(101\) 50.9117i 0.504076i 0.967717 + 0.252038i \(0.0811008\pi\)
−0.967717 + 0.252038i \(0.918899\pi\)
\(102\) 0 0
\(103\) − 110.000i − 1.06796i −0.845497 0.533981i \(-0.820696\pi\)
0.845497 0.533981i \(-0.179304\pi\)
\(104\) 8.48528i 0.0815892i
\(105\) 0 0
\(106\) 136.000 1.28302
\(107\) 118.794 1.11022 0.555112 0.831776i \(-0.312675\pi\)
0.555112 + 0.831776i \(0.312675\pi\)
\(108\) 0 0
\(109\) 25.0000 0.229358 0.114679 0.993403i \(-0.463416\pi\)
0.114679 + 0.993403i \(0.463416\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 36.0000i 0.321429i
\(113\) 45.2548 0.400485 0.200243 0.979746i \(-0.435827\pi\)
0.200243 + 0.979746i \(0.435827\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 53.7401i 0.463277i
\(117\) 0 0
\(118\) 14.0000i 0.118644i
\(119\) − 241.831i − 2.03219i
\(120\) 0 0
\(121\) −217.000 −1.79339
\(122\) 103.238 0.846210
\(123\) 0 0
\(124\) 78.0000 0.629032
\(125\) 0 0
\(126\) 0 0
\(127\) − 80.0000i − 0.629921i −0.949105 0.314961i \(-0.898009\pi\)
0.949105 0.314961i \(-0.101991\pi\)
\(128\) −33.9411 −0.265165
\(129\) 0 0
\(130\) 0 0
\(131\) − 84.8528i − 0.647731i −0.946103 0.323866i \(-0.895017\pi\)
0.946103 0.323866i \(-0.104983\pi\)
\(132\) 0 0
\(133\) 225.000i 1.69173i
\(134\) 89.0955i 0.664891i
\(135\) 0 0
\(136\) −228.000 −1.67647
\(137\) −21.2132 −0.154841 −0.0774205 0.996999i \(-0.524668\pi\)
−0.0774205 + 0.996999i \(0.524668\pi\)
\(138\) 0 0
\(139\) −104.000 −0.748201 −0.374101 0.927388i \(-0.622049\pi\)
−0.374101 + 0.927388i \(0.622049\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 88.0000i − 0.619718i
\(143\) −18.3848 −0.128565
\(144\) 0 0
\(145\) 0 0
\(146\) − 192.333i − 1.31735i
\(147\) 0 0
\(148\) − 64.0000i − 0.432432i
\(149\) − 171.120i − 1.14846i −0.818696 0.574228i \(-0.805302\pi\)
0.818696 0.574228i \(-0.194698\pi\)
\(150\) 0 0
\(151\) −17.0000 −0.112583 −0.0562914 0.998414i \(-0.517928\pi\)
−0.0562914 + 0.998414i \(0.517928\pi\)
\(152\) 212.132 1.39561
\(153\) 0 0
\(154\) −234.000 −1.51948
\(155\) 0 0
\(156\) 0 0
\(157\) 65.0000i 0.414013i 0.978340 + 0.207006i \(0.0663721\pi\)
−0.978340 + 0.207006i \(0.933628\pi\)
\(158\) 33.9411 0.214817
\(159\) 0 0
\(160\) 0 0
\(161\) − 38.1838i − 0.237166i
\(162\) 0 0
\(163\) − 17.0000i − 0.104294i −0.998639 0.0521472i \(-0.983393\pi\)
0.998639 0.0521472i \(-0.0166065\pi\)
\(164\) − 11.3137i − 0.0689860i
\(165\) 0 0
\(166\) 66.0000 0.397590
\(167\) 254.558 1.52430 0.762151 0.647399i \(-0.224143\pi\)
0.762151 + 0.647399i \(0.224143\pi\)
\(168\) 0 0
\(169\) 168.000 0.994083
\(170\) 0 0
\(171\) 0 0
\(172\) − 46.0000i − 0.267442i
\(173\) −173.948 −1.00548 −0.502741 0.864437i \(-0.667675\pi\)
−0.502741 + 0.864437i \(0.667675\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 73.5391i 0.417836i
\(177\) 0 0
\(178\) 144.000i 0.808989i
\(179\) − 125.865i − 0.703156i −0.936159 0.351578i \(-0.885645\pi\)
0.936159 0.351578i \(-0.114355\pi\)
\(180\) 0 0
\(181\) 119.000 0.657459 0.328729 0.944424i \(-0.393380\pi\)
0.328729 + 0.944424i \(0.393380\pi\)
\(182\) −12.7279 −0.0699336
\(183\) 0 0
\(184\) −36.0000 −0.195652
\(185\) 0 0
\(186\) 0 0
\(187\) − 494.000i − 2.64171i
\(188\) 65.0538 0.346031
\(189\) 0 0
\(190\) 0 0
\(191\) − 15.5563i − 0.0814469i −0.999170 0.0407234i \(-0.987034\pi\)
0.999170 0.0407234i \(-0.0129663\pi\)
\(192\) 0 0
\(193\) − 281.000i − 1.45596i −0.685599 0.727979i \(-0.740460\pi\)
0.685599 0.727979i \(-0.259540\pi\)
\(194\) − 9.89949i − 0.0510283i
\(195\) 0 0
\(196\) 64.0000 0.326531
\(197\) −224.860 −1.14142 −0.570711 0.821151i \(-0.693332\pi\)
−0.570711 + 0.821151i \(0.693332\pi\)
\(198\) 0 0
\(199\) −175.000 −0.879397 −0.439698 0.898145i \(-0.644915\pi\)
−0.439698 + 0.898145i \(0.644915\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 72.0000i 0.356436i
\(203\) −241.831 −1.19128
\(204\) 0 0
\(205\) 0 0
\(206\) − 155.563i − 0.755163i
\(207\) 0 0
\(208\) 4.00000i 0.0192308i
\(209\) 459.619i 2.19914i
\(210\) 0 0
\(211\) −49.0000 −0.232227 −0.116114 0.993236i \(-0.537044\pi\)
−0.116114 + 0.993236i \(0.537044\pi\)
\(212\) −192.333 −0.907231
\(213\) 0 0
\(214\) 168.000 0.785047
\(215\) 0 0
\(216\) 0 0
\(217\) 351.000i 1.61751i
\(218\) 35.3553 0.162180
\(219\) 0 0
\(220\) 0 0
\(221\) − 26.8701i − 0.121584i
\(222\) 0 0
\(223\) 265.000i 1.18834i 0.804339 + 0.594170i \(0.202520\pi\)
−0.804339 + 0.594170i \(0.797480\pi\)
\(224\) − 254.558i − 1.13642i
\(225\) 0 0
\(226\) 64.0000 0.283186
\(227\) 130.108 0.573161 0.286581 0.958056i \(-0.407481\pi\)
0.286581 + 0.958056i \(0.407481\pi\)
\(228\) 0 0
\(229\) 87.0000 0.379913 0.189956 0.981793i \(-0.439165\pi\)
0.189956 + 0.981793i \(0.439165\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 228.000i 0.982759i
\(233\) 152.735 0.655515 0.327758 0.944762i \(-0.393707\pi\)
0.327758 + 0.944762i \(0.393707\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) − 19.7990i − 0.0838940i
\(237\) 0 0
\(238\) − 342.000i − 1.43697i
\(239\) 277.186i 1.15977i 0.814697 + 0.579887i \(0.196903\pi\)
−0.814697 + 0.579887i \(0.803097\pi\)
\(240\) 0 0
\(241\) 415.000 1.72199 0.860996 0.508612i \(-0.169841\pi\)
0.860996 + 0.508612i \(0.169841\pi\)
\(242\) −306.884 −1.26812
\(243\) 0 0
\(244\) −146.000 −0.598361
\(245\) 0 0
\(246\) 0 0
\(247\) 25.0000i 0.101215i
\(248\) 330.926 1.33438
\(249\) 0 0
\(250\) 0 0
\(251\) 373.352i 1.48746i 0.668480 + 0.743730i \(0.266945\pi\)
−0.668480 + 0.743730i \(0.733055\pi\)
\(252\) 0 0
\(253\) − 78.0000i − 0.308300i
\(254\) − 113.137i − 0.445422i
\(255\) 0 0
\(256\) −272.000 −1.06250
\(257\) −271.529 −1.05653 −0.528267 0.849079i \(-0.677158\pi\)
−0.528267 + 0.849079i \(0.677158\pi\)
\(258\) 0 0
\(259\) 288.000 1.11197
\(260\) 0 0
\(261\) 0 0
\(262\) − 120.000i − 0.458015i
\(263\) 209.304 0.795831 0.397916 0.917422i \(-0.369734\pi\)
0.397916 + 0.917422i \(0.369734\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 318.198i 1.19623i
\(267\) 0 0
\(268\) − 126.000i − 0.470149i
\(269\) 94.7523i 0.352239i 0.984369 + 0.176120i \(0.0563545\pi\)
−0.984369 + 0.176120i \(0.943645\pi\)
\(270\) 0 0
\(271\) −176.000 −0.649446 −0.324723 0.945809i \(-0.605271\pi\)
−0.324723 + 0.945809i \(0.605271\pi\)
\(272\) −107.480 −0.395148
\(273\) 0 0
\(274\) −30.0000 −0.109489
\(275\) 0 0
\(276\) 0 0
\(277\) 303.000i 1.09386i 0.837177 + 0.546931i \(0.184204\pi\)
−0.837177 + 0.546931i \(0.815796\pi\)
\(278\) −147.078 −0.529058
\(279\) 0 0
\(280\) 0 0
\(281\) 513.360i 1.82690i 0.406948 + 0.913451i \(0.366593\pi\)
−0.406948 + 0.913451i \(0.633407\pi\)
\(282\) 0 0
\(283\) 393.000i 1.38869i 0.719641 + 0.694346i \(0.244306\pi\)
−0.719641 + 0.694346i \(0.755694\pi\)
\(284\) 124.451i 0.438207i
\(285\) 0 0
\(286\) −26.0000 −0.0909091
\(287\) 50.9117 0.177393
\(288\) 0 0
\(289\) 433.000 1.49827
\(290\) 0 0
\(291\) 0 0
\(292\) 272.000i 0.931507i
\(293\) −46.6690 −0.159280 −0.0796400 0.996824i \(-0.525377\pi\)
−0.0796400 + 0.996824i \(0.525377\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) − 271.529i − 0.917328i
\(297\) 0 0
\(298\) − 242.000i − 0.812081i
\(299\) − 4.24264i − 0.0141894i
\(300\) 0 0
\(301\) 207.000 0.687708
\(302\) −24.0416 −0.0796080
\(303\) 0 0
\(304\) 100.000 0.328947
\(305\) 0 0
\(306\) 0 0
\(307\) 289.000i 0.941368i 0.882302 + 0.470684i \(0.155993\pi\)
−0.882302 + 0.470684i \(0.844007\pi\)
\(308\) 330.926 1.07443
\(309\) 0 0
\(310\) 0 0
\(311\) − 462.448i − 1.48697i −0.668752 0.743485i \(-0.733171\pi\)
0.668752 0.743485i \(-0.266829\pi\)
\(312\) 0 0
\(313\) 481.000i 1.53674i 0.640005 + 0.768371i \(0.278932\pi\)
−0.640005 + 0.768371i \(0.721068\pi\)
\(314\) 91.9239i 0.292751i
\(315\) 0 0
\(316\) −48.0000 −0.151899
\(317\) 11.3137 0.0356899 0.0178450 0.999841i \(-0.494319\pi\)
0.0178450 + 0.999841i \(0.494319\pi\)
\(318\) 0 0
\(319\) −494.000 −1.54859
\(320\) 0 0
\(321\) 0 0
\(322\) − 54.0000i − 0.167702i
\(323\) −671.751 −2.07973
\(324\) 0 0
\(325\) 0 0
\(326\) − 24.0416i − 0.0737473i
\(327\) 0 0
\(328\) − 48.0000i − 0.146341i
\(329\) 292.742i 0.889794i
\(330\) 0 0
\(331\) −72.0000 −0.217523 −0.108761 0.994068i \(-0.534688\pi\)
−0.108761 + 0.994068i \(0.534688\pi\)
\(332\) −93.3381 −0.281139
\(333\) 0 0
\(334\) 360.000 1.07784
\(335\) 0 0
\(336\) 0 0
\(337\) − 335.000i − 0.994065i −0.867732 0.497033i \(-0.834423\pi\)
0.867732 0.497033i \(-0.165577\pi\)
\(338\) 237.588 0.702923
\(339\) 0 0
\(340\) 0 0
\(341\) 717.006i 2.10266i
\(342\) 0 0
\(343\) − 153.000i − 0.446064i
\(344\) − 195.161i − 0.567330i
\(345\) 0 0
\(346\) −246.000 −0.710983
\(347\) 383.252 1.10447 0.552236 0.833688i \(-0.313775\pi\)
0.552236 + 0.833688i \(0.313775\pi\)
\(348\) 0 0
\(349\) −568.000 −1.62751 −0.813754 0.581210i \(-0.802579\pi\)
−0.813754 + 0.581210i \(0.802579\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 520.000i − 1.47727i
\(353\) 131.522 0.372583 0.186292 0.982495i \(-0.440353\pi\)
0.186292 + 0.982495i \(0.440353\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) − 203.647i − 0.572041i
\(357\) 0 0
\(358\) − 178.000i − 0.497207i
\(359\) − 424.264i − 1.18179i −0.806747 0.590897i \(-0.798774\pi\)
0.806747 0.590897i \(-0.201226\pi\)
\(360\) 0 0
\(361\) 264.000 0.731302
\(362\) 168.291 0.464893
\(363\) 0 0
\(364\) 18.0000 0.0494505
\(365\) 0 0
\(366\) 0 0
\(367\) − 369.000i − 1.00545i −0.864447 0.502725i \(-0.832331\pi\)
0.864447 0.502725i \(-0.167669\pi\)
\(368\) −16.9706 −0.0461157
\(369\) 0 0
\(370\) 0 0
\(371\) − 865.499i − 2.33288i
\(372\) 0 0
\(373\) − 313.000i − 0.839142i −0.907723 0.419571i \(-0.862181\pi\)
0.907723 0.419571i \(-0.137819\pi\)
\(374\) − 698.621i − 1.86797i
\(375\) 0 0
\(376\) 276.000 0.734043
\(377\) −26.8701 −0.0712734
\(378\) 0 0
\(379\) 47.0000 0.124011 0.0620053 0.998076i \(-0.480250\pi\)
0.0620053 + 0.998076i \(0.480250\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 22.0000i − 0.0575916i
\(383\) −373.352 −0.974810 −0.487405 0.873176i \(-0.662057\pi\)
−0.487405 + 0.873176i \(0.662057\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 397.394i − 1.02952i
\(387\) 0 0
\(388\) 14.0000i 0.0360825i
\(389\) 390.323i 1.00340i 0.865041 + 0.501700i \(0.167292\pi\)
−0.865041 + 0.501700i \(0.832708\pi\)
\(390\) 0 0
\(391\) 114.000 0.291560
\(392\) 271.529 0.692676
\(393\) 0 0
\(394\) −318.000 −0.807107
\(395\) 0 0
\(396\) 0 0
\(397\) − 47.0000i − 0.118388i −0.998247 0.0591940i \(-0.981147\pi\)
0.998247 0.0591940i \(-0.0188530\pi\)
\(398\) −247.487 −0.621828
\(399\) 0 0
\(400\) 0 0
\(401\) − 152.735i − 0.380885i −0.981698 0.190443i \(-0.939008\pi\)
0.981698 0.190443i \(-0.0609923\pi\)
\(402\) 0 0
\(403\) 39.0000i 0.0967742i
\(404\) − 101.823i − 0.252038i
\(405\) 0 0
\(406\) −342.000 −0.842365
\(407\) 588.313 1.44549
\(408\) 0 0
\(409\) 103.000 0.251834 0.125917 0.992041i \(-0.459813\pi\)
0.125917 + 0.992041i \(0.459813\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 220.000i 0.533981i
\(413\) 89.0955 0.215727
\(414\) 0 0
\(415\) 0 0
\(416\) − 28.2843i − 0.0679910i
\(417\) 0 0
\(418\) 650.000i 1.55502i
\(419\) − 39.5980i − 0.0945059i −0.998883 0.0472530i \(-0.984953\pi\)
0.998883 0.0472530i \(-0.0150467\pi\)
\(420\) 0 0
\(421\) −560.000 −1.33017 −0.665083 0.746769i \(-0.731604\pi\)
−0.665083 + 0.746769i \(0.731604\pi\)
\(422\) −69.2965 −0.164210
\(423\) 0 0
\(424\) −816.000 −1.92453
\(425\) 0 0
\(426\) 0 0
\(427\) − 657.000i − 1.53864i
\(428\) −237.588 −0.555112
\(429\) 0 0
\(430\) 0 0
\(431\) − 479.418i − 1.11234i −0.831069 0.556170i \(-0.812270\pi\)
0.831069 0.556170i \(-0.187730\pi\)
\(432\) 0 0
\(433\) 257.000i 0.593533i 0.954950 + 0.296767i \(0.0959084\pi\)
−0.954950 + 0.296767i \(0.904092\pi\)
\(434\) 496.389i 1.14375i
\(435\) 0 0
\(436\) −50.0000 −0.114679
\(437\) −106.066 −0.242714
\(438\) 0 0
\(439\) 167.000 0.380410 0.190205 0.981744i \(-0.439085\pi\)
0.190205 + 0.981744i \(0.439085\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 38.0000i − 0.0859729i
\(443\) −169.706 −0.383083 −0.191541 0.981485i \(-0.561349\pi\)
−0.191541 + 0.981485i \(0.561349\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 374.767i 0.840284i
\(447\) 0 0
\(448\) − 504.000i − 1.12500i
\(449\) − 690.136i − 1.53705i −0.639819 0.768526i \(-0.720991\pi\)
0.639819 0.768526i \(-0.279009\pi\)
\(450\) 0 0
\(451\) 104.000 0.230599
\(452\) −90.5097 −0.200243
\(453\) 0 0
\(454\) 184.000 0.405286
\(455\) 0 0
\(456\) 0 0
\(457\) − 50.0000i − 0.109409i −0.998503 0.0547046i \(-0.982578\pi\)
0.998503 0.0547046i \(-0.0174217\pi\)
\(458\) 123.037 0.268639
\(459\) 0 0
\(460\) 0 0
\(461\) 152.735i 0.331313i 0.986184 + 0.165656i \(0.0529742\pi\)
−0.986184 + 0.165656i \(0.947026\pi\)
\(462\) 0 0
\(463\) 240.000i 0.518359i 0.965829 + 0.259179i \(0.0834520\pi\)
−0.965829 + 0.259179i \(0.916548\pi\)
\(464\) 107.480i 0.231638i
\(465\) 0 0
\(466\) 216.000 0.463519
\(467\) 592.555 1.26886 0.634428 0.772982i \(-0.281236\pi\)
0.634428 + 0.772982i \(0.281236\pi\)
\(468\) 0 0
\(469\) 567.000 1.20896
\(470\) 0 0
\(471\) 0 0
\(472\) − 84.0000i − 0.177966i
\(473\) 422.850 0.893974
\(474\) 0 0
\(475\) 0 0
\(476\) 483.661i 1.01609i
\(477\) 0 0
\(478\) 392.000i 0.820084i
\(479\) − 496.389i − 1.03630i −0.855289 0.518151i \(-0.826620\pi\)
0.855289 0.518151i \(-0.173380\pi\)
\(480\) 0 0
\(481\) 32.0000 0.0665281
\(482\) 586.899 1.21763
\(483\) 0 0
\(484\) 434.000 0.896694
\(485\) 0 0
\(486\) 0 0
\(487\) 545.000i 1.11910i 0.828798 + 0.559548i \(0.189025\pi\)
−0.828798 + 0.559548i \(0.810975\pi\)
\(488\) −619.426 −1.26931
\(489\) 0 0
\(490\) 0 0
\(491\) 905.097i 1.84337i 0.387934 + 0.921687i \(0.373189\pi\)
−0.387934 + 0.921687i \(0.626811\pi\)
\(492\) 0 0
\(493\) − 722.000i − 1.46450i
\(494\) 35.3553i 0.0715695i
\(495\) 0 0
\(496\) 156.000 0.314516
\(497\) −560.029 −1.12682
\(498\) 0 0
\(499\) −7.00000 −0.0140281 −0.00701403 0.999975i \(-0.502233\pi\)
−0.00701403 + 0.999975i \(0.502233\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 528.000i 1.05179i
\(503\) 786.303 1.56323 0.781613 0.623764i \(-0.214397\pi\)
0.781613 + 0.623764i \(0.214397\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) − 110.309i − 0.218001i
\(507\) 0 0
\(508\) 160.000i 0.314961i
\(509\) 578.413i 1.13637i 0.822900 + 0.568186i \(0.192355\pi\)
−0.822900 + 0.568186i \(0.807645\pi\)
\(510\) 0 0
\(511\) −1224.00 −2.39530
\(512\) −248.902 −0.486136
\(513\) 0 0
\(514\) −384.000 −0.747082
\(515\) 0 0
\(516\) 0 0
\(517\) 598.000i 1.15667i
\(518\) 407.294 0.786281
\(519\) 0 0
\(520\) 0 0
\(521\) − 414.365i − 0.795325i −0.917532 0.397663i \(-0.869821\pi\)
0.917532 0.397663i \(-0.130179\pi\)
\(522\) 0 0
\(523\) 263.000i 0.502868i 0.967874 + 0.251434i \(0.0809022\pi\)
−0.967874 + 0.251434i \(0.919098\pi\)
\(524\) 169.706i 0.323866i
\(525\) 0 0
\(526\) 296.000 0.562738
\(527\) −1047.93 −1.98849
\(528\) 0 0
\(529\) −511.000 −0.965974
\(530\) 0 0
\(531\) 0 0
\(532\) − 450.000i − 0.845865i
\(533\) 5.65685 0.0106132
\(534\) 0 0
\(535\) 0 0
\(536\) − 534.573i − 0.997337i
\(537\) 0 0
\(538\) 134.000i 0.249071i
\(539\) 588.313i 1.09149i
\(540\) 0 0
\(541\) 489.000 0.903882 0.451941 0.892048i \(-0.350732\pi\)
0.451941 + 0.892048i \(0.350732\pi\)
\(542\) −248.902 −0.459228
\(543\) 0 0
\(544\) 760.000 1.39706
\(545\) 0 0
\(546\) 0 0
\(547\) 10.0000i 0.0182815i 0.999958 + 0.00914077i \(0.00290964\pi\)
−0.999958 + 0.00914077i \(0.997090\pi\)
\(548\) 42.4264 0.0774205
\(549\) 0 0
\(550\) 0 0
\(551\) 671.751i 1.21915i
\(552\) 0 0
\(553\) − 216.000i − 0.390597i
\(554\) 428.507i 0.773478i
\(555\) 0 0
\(556\) 208.000 0.374101
\(557\) −623.668 −1.11969 −0.559846 0.828597i \(-0.689140\pi\)
−0.559846 + 0.828597i \(0.689140\pi\)
\(558\) 0 0
\(559\) 23.0000 0.0411449
\(560\) 0 0
\(561\) 0 0
\(562\) 726.000i 1.29181i
\(563\) −63.6396 −0.113037 −0.0565183 0.998402i \(-0.518000\pi\)
−0.0565183 + 0.998402i \(0.518000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 555.786i 0.981954i
\(567\) 0 0
\(568\) 528.000i 0.929577i
\(569\) − 810.344i − 1.42416i −0.702101 0.712078i \(-0.747754\pi\)
0.702101 0.712078i \(-0.252246\pi\)
\(570\) 0 0
\(571\) 377.000 0.660245 0.330123 0.943938i \(-0.392910\pi\)
0.330123 + 0.943938i \(0.392910\pi\)
\(572\) 36.7696 0.0642824
\(573\) 0 0
\(574\) 72.0000 0.125436
\(575\) 0 0
\(576\) 0 0
\(577\) 791.000i 1.37088i 0.728127 + 0.685442i \(0.240391\pi\)
−0.728127 + 0.685442i \(0.759609\pi\)
\(578\) 612.354 1.05944
\(579\) 0 0
\(580\) 0 0
\(581\) − 420.021i − 0.722928i
\(582\) 0 0
\(583\) − 1768.00i − 3.03259i
\(584\) 1154.00i 1.97602i
\(585\) 0 0
\(586\) −66.0000 −0.112628
\(587\) −356.382 −0.607124 −0.303562 0.952812i \(-0.598176\pi\)
−0.303562 + 0.952812i \(0.598176\pi\)
\(588\) 0 0
\(589\) 975.000 1.65535
\(590\) 0 0
\(591\) 0 0
\(592\) − 128.000i − 0.216216i
\(593\) −435.578 −0.734533 −0.367266 0.930116i \(-0.619706\pi\)
−0.367266 + 0.930116i \(0.619706\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 342.240i 0.574228i
\(597\) 0 0
\(598\) − 6.00000i − 0.0100334i
\(599\) − 1108.74i − 1.85099i −0.378759 0.925495i \(-0.623649\pi\)
0.378759 0.925495i \(-0.376351\pi\)
\(600\) 0 0
\(601\) 233.000 0.387687 0.193844 0.981032i \(-0.437905\pi\)
0.193844 + 0.981032i \(0.437905\pi\)
\(602\) 292.742 0.486283
\(603\) 0 0
\(604\) 34.0000 0.0562914
\(605\) 0 0
\(606\) 0 0
\(607\) − 776.000i − 1.27842i −0.769033 0.639209i \(-0.779262\pi\)
0.769033 0.639209i \(-0.220738\pi\)
\(608\) −707.107 −1.16300
\(609\) 0 0
\(610\) 0 0
\(611\) 32.5269i 0.0532355i
\(612\) 0 0
\(613\) − 376.000i − 0.613377i −0.951810 0.306688i \(-0.900779\pi\)
0.951810 0.306688i \(-0.0992210\pi\)
\(614\) 408.708i 0.665648i
\(615\) 0 0
\(616\) 1404.00 2.27922
\(617\) −593.970 −0.962674 −0.481337 0.876536i \(-0.659849\pi\)
−0.481337 + 0.876536i \(0.659849\pi\)
\(618\) 0 0
\(619\) −1031.00 −1.66559 −0.832795 0.553582i \(-0.813261\pi\)
−0.832795 + 0.553582i \(0.813261\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 654.000i − 1.05145i
\(623\) 916.410 1.47096
\(624\) 0 0
\(625\) 0 0
\(626\) 680.237i 1.08664i
\(627\) 0 0
\(628\) − 130.000i − 0.207006i
\(629\) 859.842i 1.36700i
\(630\) 0 0
\(631\) 745.000 1.18067 0.590333 0.807160i \(-0.298997\pi\)
0.590333 + 0.807160i \(0.298997\pi\)
\(632\) −203.647 −0.322226
\(633\) 0 0
\(634\) 16.0000 0.0252366
\(635\) 0 0
\(636\) 0 0
\(637\) 32.0000i 0.0502355i
\(638\) −698.621 −1.09502
\(639\) 0 0
\(640\) 0 0
\(641\) − 203.647i − 0.317702i −0.987303 0.158851i \(-0.949221\pi\)
0.987303 0.158851i \(-0.0507789\pi\)
\(642\) 0 0
\(643\) 1032.00i 1.60498i 0.596668 + 0.802488i \(0.296491\pi\)
−0.596668 + 0.802488i \(0.703509\pi\)
\(644\) 76.3675i 0.118583i
\(645\) 0 0
\(646\) −950.000 −1.47059
\(647\) 56.5685 0.0874321 0.0437160 0.999044i \(-0.486080\pi\)
0.0437160 + 0.999044i \(0.486080\pi\)
\(648\) 0 0
\(649\) 182.000 0.280431
\(650\) 0 0
\(651\) 0 0
\(652\) 34.0000i 0.0521472i
\(653\) 272.943 0.417983 0.208992 0.977917i \(-0.432982\pi\)
0.208992 + 0.977917i \(0.432982\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) − 22.6274i − 0.0344930i
\(657\) 0 0
\(658\) 414.000i 0.629179i
\(659\) 379.009i 0.575128i 0.957761 + 0.287564i \(0.0928454\pi\)
−0.957761 + 0.287564i \(0.907155\pi\)
\(660\) 0 0
\(661\) −440.000 −0.665658 −0.332829 0.942987i \(-0.608003\pi\)
−0.332829 + 0.942987i \(0.608003\pi\)
\(662\) −101.823 −0.153812
\(663\) 0 0
\(664\) −396.000 −0.596386
\(665\) 0 0
\(666\) 0 0
\(667\) − 114.000i − 0.170915i
\(668\) −509.117 −0.762151
\(669\) 0 0
\(670\) 0 0
\(671\) − 1342.09i − 2.00013i
\(672\) 0 0
\(673\) 264.000i 0.392273i 0.980577 + 0.196137i \(0.0628396\pi\)
−0.980577 + 0.196137i \(0.937160\pi\)
\(674\) − 473.762i − 0.702910i
\(675\) 0 0
\(676\) −336.000 −0.497041
\(677\) 538.815 0.795887 0.397943 0.917410i \(-0.369724\pi\)
0.397943 + 0.917410i \(0.369724\pi\)
\(678\) 0 0
\(679\) −63.0000 −0.0927835
\(680\) 0 0
\(681\) 0 0
\(682\) 1014.00i 1.48680i
\(683\) 576.999 0.844801 0.422401 0.906409i \(-0.361188\pi\)
0.422401 + 0.906409i \(0.361188\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) − 216.375i − 0.315415i
\(687\) 0 0
\(688\) − 92.0000i − 0.133721i
\(689\) − 96.1665i − 0.139574i
\(690\) 0 0
\(691\) −182.000 −0.263386 −0.131693 0.991291i \(-0.542041\pi\)
−0.131693 + 0.991291i \(0.542041\pi\)
\(692\) 347.897 0.502741
\(693\) 0 0
\(694\) 542.000 0.780980
\(695\) 0 0
\(696\) 0 0
\(697\) 152.000i 0.218077i
\(698\) −803.273 −1.15082
\(699\) 0 0
\(700\) 0 0
\(701\) − 558.614i − 0.796882i −0.917194 0.398441i \(-0.869551\pi\)
0.917194 0.398441i \(-0.130449\pi\)
\(702\) 0 0
\(703\) − 800.000i − 1.13798i
\(704\) − 1029.55i − 1.46243i
\(705\) 0 0
\(706\) 186.000 0.263456
\(707\) 458.205 0.648098
\(708\) 0 0
\(709\) −607.000 −0.856135 −0.428068 0.903747i \(-0.640806\pi\)
−0.428068 + 0.903747i \(0.640806\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 864.000i − 1.21348i
\(713\) −165.463 −0.232066
\(714\) 0 0
\(715\) 0 0
\(716\) 251.730i 0.351578i
\(717\) 0 0
\(718\) − 600.000i − 0.835655i
\(719\) − 106.066i − 0.147519i −0.997276 0.0737594i \(-0.976500\pi\)
0.997276 0.0737594i \(-0.0234997\pi\)
\(720\) 0 0
\(721\) −990.000 −1.37309
\(722\) 373.352 0.517109
\(723\) 0 0
\(724\) −238.000 −0.328729
\(725\) 0 0
\(726\) 0 0
\(727\) 71.0000i 0.0976616i 0.998807 + 0.0488308i \(0.0155495\pi\)
−0.998807 + 0.0488308i \(0.984450\pi\)
\(728\) 76.3675 0.104900
\(729\) 0 0
\(730\) 0 0
\(731\) 618.011i 0.845433i
\(732\) 0 0
\(733\) 904.000i 1.23329i 0.787242 + 0.616644i \(0.211508\pi\)
−0.787242 + 0.616644i \(0.788492\pi\)
\(734\) − 521.845i − 0.710960i
\(735\) 0 0
\(736\) 120.000 0.163043
\(737\) 1158.24 1.57156
\(738\) 0 0
\(739\) 800.000 1.08254 0.541272 0.840848i \(-0.317943\pi\)
0.541272 + 0.840848i \(0.317943\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 1224.00i − 1.64960i
\(743\) 418.607 0.563401 0.281701 0.959502i \(-0.409101\pi\)
0.281701 + 0.959502i \(0.409101\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) − 442.649i − 0.593363i
\(747\) 0 0
\(748\) 988.000i 1.32086i
\(749\) − 1069.15i − 1.42743i
\(750\) 0 0
\(751\) −704.000 −0.937417 −0.468708 0.883353i \(-0.655280\pi\)
−0.468708 + 0.883353i \(0.655280\pi\)
\(752\) 130.108 0.173015
\(753\) 0 0
\(754\) −38.0000 −0.0503979
\(755\) 0 0
\(756\) 0 0
\(757\) − 977.000i − 1.29062i −0.763920 0.645310i \(-0.776728\pi\)
0.763920 0.645310i \(-0.223272\pi\)
\(758\) 66.4680 0.0876887
\(759\) 0 0
\(760\) 0 0
\(761\) 746.705i 0.981215i 0.871381 + 0.490608i \(0.163225\pi\)
−0.871381 + 0.490608i \(0.836775\pi\)
\(762\) 0 0
\(763\) − 225.000i − 0.294889i
\(764\) 31.1127i 0.0407234i
\(765\) 0 0
\(766\) −528.000 −0.689295
\(767\) 9.89949 0.0129068
\(768\) 0 0
\(769\) −1337.00 −1.73862 −0.869311 0.494266i \(-0.835437\pi\)
−0.869311 + 0.494266i \(0.835437\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 562.000i 0.727979i
\(773\) 1170.97 1.51484 0.757418 0.652930i \(-0.226460\pi\)
0.757418 + 0.652930i \(0.226460\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 59.3970i 0.0765425i
\(777\) 0 0
\(778\) 552.000i 0.709512i
\(779\) − 141.421i − 0.181542i
\(780\) 0 0
\(781\) −1144.00 −1.46479
\(782\) 161.220 0.206164
\(783\) 0 0
\(784\) 128.000 0.163265
\(785\) 0 0
\(786\) 0 0
\(787\) 703.000i 0.893266i 0.894717 + 0.446633i \(0.147377\pi\)
−0.894717 + 0.446633i \(0.852623\pi\)
\(788\) 449.720 0.570711
\(789\) 0 0
\(790\) 0 0
\(791\) − 407.294i − 0.514910i
\(792\) 0 0
\(793\) − 73.0000i − 0.0920555i
\(794\) − 66.4680i − 0.0837129i
\(795\) 0 0
\(796\) 350.000 0.439698
\(797\) 1538.66 1.93057 0.965285 0.261199i \(-0.0841178\pi\)
0.965285 + 0.261199i \(0.0841178\pi\)
\(798\) 0 0
\(799\) −874.000 −1.09387
\(800\) 0 0
\(801\) 0 0
\(802\) − 216.000i − 0.269327i
\(803\) −2500.33 −3.11374
\(804\) 0 0
\(805\) 0 0
\(806\) 55.1543i 0.0684297i
\(807\) 0 0
\(808\) − 432.000i − 0.534653i
\(809\) − 73.5391i − 0.0909012i −0.998967 0.0454506i \(-0.985528\pi\)
0.998967 0.0454506i \(-0.0144724\pi\)
\(810\) 0 0
\(811\) 1095.00 1.35018 0.675092 0.737733i \(-0.264104\pi\)
0.675092 + 0.737733i \(0.264104\pi\)
\(812\) 483.661 0.595642
\(813\) 0 0
\(814\) 832.000 1.02211
\(815\) 0 0
\(816\) 0 0
\(817\) − 575.000i − 0.703794i
\(818\) 145.664 0.178073
\(819\) 0 0
\(820\) 0 0
\(821\) − 1166.73i − 1.42110i −0.703645 0.710552i \(-0.748445\pi\)
0.703645 0.710552i \(-0.251555\pi\)
\(822\) 0 0
\(823\) 1015.00i 1.23329i 0.787240 + 0.616646i \(0.211509\pi\)
−0.787240 + 0.616646i \(0.788491\pi\)
\(824\) 933.381i 1.13274i
\(825\) 0 0
\(826\) 126.000 0.152542
\(827\) 379.009 0.458294 0.229147 0.973392i \(-0.426406\pi\)
0.229147 + 0.973392i \(0.426406\pi\)
\(828\) 0 0
\(829\) −150.000 −0.180941 −0.0904704 0.995899i \(-0.528837\pi\)
−0.0904704 + 0.995899i \(0.528837\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 56.0000i − 0.0673077i
\(833\) −859.842 −1.03222
\(834\) 0 0
\(835\) 0 0
\(836\) − 919.239i − 1.09957i
\(837\) 0 0
\(838\) − 56.0000i − 0.0668258i
\(839\) 660.438i 0.787173i 0.919288 + 0.393586i \(0.128766\pi\)
−0.919288 + 0.393586i \(0.871234\pi\)
\(840\) 0 0
\(841\) 119.000 0.141498
\(842\) −791.960 −0.940570
\(843\) 0 0
\(844\) 98.0000 0.116114
\(845\) 0 0
\(846\) 0 0
\(847\) 1953.00i 2.30579i
\(848\) −384.666 −0.453616
\(849\) 0 0
\(850\) 0 0
\(851\) 135.765i 0.159535i
\(852\) 0 0
\(853\) 297.000i 0.348183i 0.984729 + 0.174091i \(0.0556988\pi\)
−0.984729 + 0.174091i \(0.944301\pi\)
\(854\) − 929.138i − 1.08798i
\(855\) 0 0
\(856\) −1008.00 −1.17757
\(857\) −1165.31 −1.35976 −0.679879 0.733325i \(-0.737968\pi\)
−0.679879 + 0.733325i \(0.737968\pi\)
\(858\) 0 0
\(859\) 1286.00 1.49709 0.748545 0.663084i \(-0.230753\pi\)
0.748545 + 0.663084i \(0.230753\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 678.000i − 0.786543i
\(863\) 633.568 0.734146 0.367073 0.930192i \(-0.380360\pi\)
0.367073 + 0.930192i \(0.380360\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 363.453i 0.419692i
\(867\) 0 0
\(868\) − 702.000i − 0.808756i
\(869\) − 441.235i − 0.507750i
\(870\) 0 0
\(871\) 63.0000 0.0723307
\(872\) −212.132 −0.243271
\(873\) 0 0
\(874\) −150.000 −0.171625
\(875\) 0 0
\(876\) 0 0
\(877\) − 1599.00i − 1.82326i −0.411011 0.911631i \(-0.634824\pi\)
0.411011 0.911631i \(-0.365176\pi\)
\(878\) 236.174 0.268991
\(879\) 0 0
\(880\) 0 0
\(881\) 961.665i 1.09156i 0.837928 + 0.545780i \(0.183767\pi\)
−0.837928 + 0.545780i \(0.816233\pi\)
\(882\) 0 0
\(883\) 193.000i 0.218573i 0.994010 + 0.109287i \(0.0348566\pi\)
−0.994010 + 0.109287i \(0.965143\pi\)
\(884\) 53.7401i 0.0607920i
\(885\) 0 0
\(886\) −240.000 −0.270880
\(887\) −700.036 −0.789217 −0.394609 0.918849i \(-0.629120\pi\)
−0.394609 + 0.918849i \(0.629120\pi\)
\(888\) 0 0
\(889\) −720.000 −0.809899
\(890\) 0 0
\(891\) 0 0
\(892\) − 530.000i − 0.594170i
\(893\) 813.173 0.910608
\(894\) 0 0
\(895\) 0 0
\(896\) 305.470i 0.340926i
\(897\) 0 0
\(898\) − 976.000i − 1.08686i
\(899\) 1047.93i 1.16566i
\(900\) 0 0
\(901\) 2584.00 2.86792
\(902\) 147.078 0.163058
\(903\) 0 0
\(904\) −384.000 −0.424779
\(905\) 0 0
\(906\) 0 0
\(907\) − 240.000i − 0.264609i −0.991209 0.132304i \(-0.957762\pi\)
0.991209 0.132304i \(-0.0422376\pi\)
\(908\) −260.215 −0.286581
\(909\) 0 0
\(910\) 0 0
\(911\) − 171.120i − 0.187837i −0.995580 0.0939187i \(-0.970061\pi\)
0.995580 0.0939187i \(-0.0299394\pi\)
\(912\) 0 0
\(913\) − 858.000i − 0.939759i
\(914\) − 70.7107i − 0.0773640i
\(915\) 0 0
\(916\) −174.000 −0.189956
\(917\) −763.675 −0.832798
\(918\) 0 0
\(919\) −343.000 −0.373232 −0.186616 0.982433i \(-0.559752\pi\)
−0.186616 + 0.982433i \(0.559752\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 216.000i 0.234273i
\(923\) −62.2254 −0.0674165
\(924\) 0 0
\(925\) 0 0
\(926\) 339.411i 0.366535i
\(927\) 0 0
\(928\) − 760.000i − 0.818966i
\(929\) 719.835i 0.774849i 0.921901 + 0.387424i \(0.126635\pi\)
−0.921901 + 0.387424i \(0.873365\pi\)
\(930\) 0 0
\(931\) 800.000 0.859291
\(932\) −305.470 −0.327758
\(933\) 0 0
\(934\) 838.000 0.897216
\(935\) 0 0
\(936\) 0 0
\(937\) 609.000i 0.649947i 0.945723 + 0.324973i \(0.105355\pi\)
−0.945723 + 0.324973i \(0.894645\pi\)
\(938\) 801.859 0.854860
\(939\) 0 0
\(940\) 0 0
\(941\) 111.723i 0.118728i 0.998236 + 0.0593639i \(0.0189072\pi\)
−0.998236 + 0.0593639i \(0.981093\pi\)
\(942\) 0 0
\(943\) 24.0000i 0.0254507i
\(944\) − 39.5980i − 0.0419470i
\(945\) 0 0
\(946\) 598.000 0.632135
\(947\) −1022.48 −1.07970 −0.539850 0.841761i \(-0.681519\pi\)
−0.539850 + 0.841761i \(0.681519\pi\)
\(948\) 0 0
\(949\) −136.000 −0.143309
\(950\) 0 0
\(951\) 0 0
\(952\) 2052.00i 2.15546i
\(953\) −1380.27 −1.44834 −0.724172 0.689619i \(-0.757778\pi\)
−0.724172 + 0.689619i \(0.757778\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) − 554.372i − 0.579887i
\(957\) 0 0
\(958\) − 702.000i − 0.732777i
\(959\) 190.919i 0.199081i
\(960\) 0 0
\(961\) 560.000 0.582726
\(962\) 45.2548 0.0470424
\(963\) 0 0
\(964\) −830.000 −0.860996
\(965\) 0 0
\(966\) 0 0
\(967\) 584.000i 0.603930i 0.953319 + 0.301965i \(0.0976425\pi\)
−0.953319 + 0.301965i \(0.902357\pi\)
\(968\) 1841.31 1.90218
\(969\) 0 0
\(970\) 0 0
\(971\) 1030.96i 1.06175i 0.847449 + 0.530876i \(0.178137\pi\)
−0.847449 + 0.530876i \(0.821863\pi\)
\(972\) 0 0
\(973\) 936.000i 0.961973i
\(974\) 770.746i 0.791321i
\(975\) 0 0
\(976\) −292.000 −0.299180
\(977\) 1028.13 1.05234 0.526169 0.850380i \(-0.323628\pi\)
0.526169 + 0.850380i \(0.323628\pi\)
\(978\) 0 0
\(979\) 1872.00 1.91216
\(980\) 0 0
\(981\) 0 0
\(982\) 1280.00i 1.30346i
\(983\) 1323.70 1.34660 0.673298 0.739371i \(-0.264877\pi\)
0.673298 + 0.739371i \(0.264877\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) − 1021.06i − 1.03556i
\(987\) 0 0
\(988\) − 50.0000i − 0.0506073i
\(989\) 97.5807i 0.0986661i
\(990\) 0 0
\(991\) −961.000 −0.969728 −0.484864 0.874590i \(-0.661131\pi\)
−0.484864 + 0.874590i \(0.661131\pi\)
\(992\) −1103.09 −1.11198
\(993\) 0 0
\(994\) −792.000 −0.796781
\(995\) 0 0
\(996\) 0 0
\(997\) − 600.000i − 0.601805i −0.953655 0.300903i \(-0.902712\pi\)
0.953655 0.300903i \(-0.0972879\pi\)
\(998\) −9.89949 −0.00991933
\(999\) 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 225.3.d.a.224.3 4
3.2 odd 2 inner 225.3.d.a.224.1 4
4.3 odd 2 3600.3.c.e.449.4 4
5.2 odd 4 225.3.c.b.26.2 yes 2
5.3 odd 4 225.3.c.a.26.1 2
5.4 even 2 inner 225.3.d.a.224.2 4
12.11 even 2 3600.3.c.e.449.3 4
15.2 even 4 225.3.c.b.26.1 yes 2
15.8 even 4 225.3.c.a.26.2 yes 2
15.14 odd 2 inner 225.3.d.a.224.4 4
20.3 even 4 3600.3.l.j.1601.2 2
20.7 even 4 3600.3.l.b.1601.2 2
20.19 odd 2 3600.3.c.e.449.2 4
60.23 odd 4 3600.3.l.j.1601.1 2
60.47 odd 4 3600.3.l.b.1601.1 2
60.59 even 2 3600.3.c.e.449.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.3.c.a.26.1 2 5.3 odd 4
225.3.c.a.26.2 yes 2 15.8 even 4
225.3.c.b.26.1 yes 2 15.2 even 4
225.3.c.b.26.2 yes 2 5.2 odd 4
225.3.d.a.224.1 4 3.2 odd 2 inner
225.3.d.a.224.2 4 5.4 even 2 inner
225.3.d.a.224.3 4 1.1 even 1 trivial
225.3.d.a.224.4 4 15.14 odd 2 inner
3600.3.c.e.449.1 4 60.59 even 2
3600.3.c.e.449.2 4 20.19 odd 2
3600.3.c.e.449.3 4 12.11 even 2
3600.3.c.e.449.4 4 4.3 odd 2
3600.3.l.b.1601.1 2 60.47 odd 4
3600.3.l.b.1601.2 2 20.7 even 4
3600.3.l.j.1601.1 2 60.23 odd 4
3600.3.l.j.1601.2 2 20.3 even 4