Properties

Label 225.3.d.a.224.4
Level $225$
Weight $3$
Character 225.224
Analytic conductor $6.131$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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

$q$-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 \(0.222251\pi\)
−0.765986 + 0.642857i \(0.777749\pi\)
\(8\) −8.48528 −1.06066
\(9\) 0 0
\(10\) 0 0
\(11\) 18.3848i 1.67134i 0.549229 + 0.835672i \(0.314921\pi\)
−0.549229 + 0.835672i \(0.685079\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.0769231i 0.999260 + 0.0384615i \(0.0122457\pi\)
−0.999260 + 0.0384615i \(0.987754\pi\)
\(14\) 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 \(0.153326\pi\)
−0.886214 + 0.463277i \(0.846674\pi\)
\(30\) 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 \(-0.857655\pi\)
0.901666 0.432432i \(-0.142345\pi\)
\(38\) −35.3553 −0.930404
\(39\) 0 0
\(40\) 0 0
\(41\) − 5.65685i − 0.137972i −0.997618 0.0689860i \(-0.978024\pi\)
0.997618 0.0689860i \(-0.0219764\pi\)
\(42\) 0 0
\(43\) − 23.0000i − 0.534884i −0.963574 0.267442i \(-0.913822\pi\)
0.963574 0.267442i \(-0.0861783\pi\)
\(44\) − 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 \(-0.973264\pi\)
0.996475 0.0838940i \(-0.0267357\pi\)
\(60\) 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 \(-0.844200\pi\)
0.882587 0.470149i \(-0.155800\pi\)
\(68\) −53.7401 −0.790296
\(69\) 0 0
\(70\) 0 0
\(71\) 62.2254i 0.876414i 0.898874 + 0.438207i \(0.144386\pi\)
−0.898874 + 0.438207i \(0.855614\pi\)
\(72\) 0 0
\(73\) 136.000i 1.86301i 0.363724 + 0.931507i \(0.381505\pi\)
−0.363724 + 0.931507i \(0.618495\pi\)
\(74\) − 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 \(-0.806152\pi\)
0.820225 0.572041i \(-0.193848\pi\)
\(90\) 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 \(0.0114879\pi\)
−0.999349 + 0.0360825i \(0.988512\pi\)
\(98\) −45.2548 −0.461784
\(99\) 0 0
\(100\) 0 0
\(101\) − 50.9117i − 0.504076i −0.967717 0.252038i \(-0.918899\pi\)
0.967717 0.252038i \(-0.0811008\pi\)
\(102\) 0 0
\(103\) 110.000i 1.06796i 0.845497 + 0.533981i \(0.179304\pi\)
−0.845497 + 0.533981i \(0.820696\pi\)
\(104\) − 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 \(0.101991\pi\)
−0.949105 + 0.314961i \(0.898009\pi\)
\(128\) −33.9411 −0.265165
\(129\) 0 0
\(130\) 0 0
\(131\) 84.8528i 0.647731i 0.946103 + 0.323866i \(0.104983\pi\)
−0.946103 + 0.323866i \(0.895017\pi\)
\(132\) 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 \(0.194698\pi\)
−0.818696 + 0.574228i \(0.805302\pi\)
\(150\) 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 \(-0.933628\pi\)
0.978340 0.207006i \(-0.0663721\pi\)
\(158\) 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 \(0.0166065\pi\)
−0.998639 + 0.0521472i \(0.983393\pi\)
\(164\) 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 \(0.114355\pi\)
−0.936159 + 0.351578i \(0.885645\pi\)
\(180\) 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 \(0.0129663\pi\)
−0.999170 + 0.0407234i \(0.987034\pi\)
\(192\) 0 0
\(193\) 281.000i 1.45596i 0.685599 + 0.727979i \(0.259540\pi\)
−0.685599 + 0.727979i \(0.740460\pi\)
\(194\) 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 \(-0.797480\pi\)
0.804339 0.594170i \(-0.202520\pi\)
\(224\) 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 \(-0.803097\pi\)
0.814697 0.579887i \(-0.196903\pi\)
\(240\) 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 \(-0.733055\pi\)
0.668480 0.743730i \(-0.266945\pi\)
\(252\) 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 \(-0.943645\pi\)
0.984369 0.176120i \(-0.0563545\pi\)
\(270\) 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 \(-0.815796\pi\)
0.837177 0.546931i \(-0.184204\pi\)
\(278\) −147.078 −0.529058
\(279\) 0 0
\(280\) 0 0
\(281\) − 513.360i − 1.82690i −0.406948 0.913451i \(-0.633407\pi\)
0.406948 0.913451i \(-0.366593\pi\)
\(282\) 0 0
\(283\) − 393.000i − 1.38869i −0.719641 0.694346i \(-0.755694\pi\)
0.719641 0.694346i \(-0.244306\pi\)
\(284\) − 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 \(-0.844007\pi\)
0.882302 0.470684i \(-0.155993\pi\)
\(308\) 330.926 1.07443
\(309\) 0 0
\(310\) 0 0
\(311\) 462.448i 1.48697i 0.668752 + 0.743485i \(0.266829\pi\)
−0.668752 + 0.743485i \(0.733171\pi\)
\(312\) 0 0
\(313\) − 481.000i − 1.53674i −0.640005 0.768371i \(-0.721068\pi\)
0.640005 0.768371i \(-0.278932\pi\)
\(314\) − 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 \(0.165577\pi\)
−0.867732 + 0.497033i \(0.834423\pi\)
\(338\) 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 \(0.201226\pi\)
−0.806747 + 0.590897i \(0.798774\pi\)
\(360\) 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 \(0.167669\pi\)
−0.864447 + 0.502725i \(0.832331\pi\)
\(368\) −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 \(0.137819\pi\)
−0.907723 + 0.419571i \(0.862181\pi\)
\(374\) 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 \(-0.832708\pi\)
0.865041 0.501700i \(-0.167292\pi\)
\(390\) 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 \(0.0188530\pi\)
−0.998247 + 0.0591940i \(0.981147\pi\)
\(398\) −247.487 −0.621828
\(399\) 0 0
\(400\) 0 0
\(401\) 152.735i 0.380885i 0.981698 + 0.190443i \(0.0609923\pi\)
−0.981698 + 0.190443i \(0.939008\pi\)
\(402\) 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 \(0.0150467\pi\)
−0.998883 + 0.0472530i \(0.984953\pi\)
\(420\) 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 \(0.187730\pi\)
−0.831069 + 0.556170i \(0.812270\pi\)
\(432\) 0 0
\(433\) − 257.000i − 0.593533i −0.954950 0.296767i \(-0.904092\pi\)
0.954950 0.296767i \(-0.0959084\pi\)
\(434\) − 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 \(0.279009\pi\)
−0.639819 + 0.768526i \(0.720991\pi\)
\(450\) 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 \(0.0174217\pi\)
−0.998503 + 0.0547046i \(0.982578\pi\)
\(458\) 123.037 0.268639
\(459\) 0 0
\(460\) 0 0
\(461\) − 152.735i − 0.331313i −0.986184 0.165656i \(-0.947026\pi\)
0.986184 0.165656i \(-0.0529742\pi\)
\(462\) 0 0
\(463\) − 240.000i − 0.518359i −0.965829 0.259179i \(-0.916548\pi\)
0.965829 0.259179i \(-0.0834520\pi\)
\(464\) − 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 \(0.173380\pi\)
−0.855289 + 0.518151i \(0.826620\pi\)
\(480\) 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 \(-0.810975\pi\)
0.828798 0.559548i \(-0.189025\pi\)
\(488\) −619.426 −1.26931
\(489\) 0 0
\(490\) 0 0
\(491\) − 905.097i − 1.84337i −0.387934 0.921687i \(-0.626811\pi\)
0.387934 0.921687i \(-0.373189\pi\)
\(492\) 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 \(-0.807645\pi\)
0.822900 0.568186i \(-0.192355\pi\)
\(510\) 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 \(0.130179\pi\)
−0.917532 + 0.397663i \(0.869821\pi\)
\(522\) 0 0
\(523\) − 263.000i − 0.502868i −0.967874 0.251434i \(-0.919098\pi\)
0.967874 0.251434i \(-0.0809022\pi\)
\(524\) − 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 \(-0.997090\pi\)
0.999958 0.00914077i \(-0.00290964\pi\)
\(548\) 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 \(0.252246\pi\)
−0.702101 + 0.712078i \(0.747754\pi\)
\(570\) 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 \(-0.759609\pi\)
0.728127 0.685442i \(-0.240391\pi\)
\(578\) 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 \(0.376351\pi\)
−0.378759 + 0.925495i \(0.623649\pi\)
\(600\) 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 \(0.220738\pi\)
−0.769033 + 0.639209i \(0.779262\pi\)
\(608\) −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 \(0.0992210\pi\)
−0.951810 + 0.306688i \(0.900779\pi\)
\(614\) − 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 \(0.0507789\pi\)
−0.987303 + 0.158851i \(0.949221\pi\)
\(642\) 0 0
\(643\) − 1032.00i − 1.60498i −0.596668 0.802488i \(-0.703509\pi\)
0.596668 0.802488i \(-0.296491\pi\)
\(644\) − 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 \(-0.907155\pi\)
0.957761 0.287564i \(-0.0928454\pi\)
\(660\) 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 \(-0.937160\pi\)
0.980577 0.196137i \(-0.0628396\pi\)
\(674\) 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 \(0.130449\pi\)
−0.917194 + 0.398441i \(0.869551\pi\)
\(702\) 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 \(0.0234997\pi\)
−0.997276 + 0.0737594i \(0.976500\pi\)
\(720\) 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 \(-0.984450\pi\)
0.998807 0.0488308i \(-0.0155495\pi\)
\(728\) 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 \(-0.788492\pi\)
0.787242 0.616644i \(-0.211508\pi\)
\(734\) 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 \(0.223272\pi\)
−0.763920 + 0.645310i \(0.776728\pi\)
\(758\) 66.4680 0.0876887
\(759\) 0 0
\(760\) 0 0
\(761\) − 746.705i − 0.981215i −0.871381 0.490608i \(-0.836775\pi\)
0.871381 0.490608i \(-0.163225\pi\)
\(762\) 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 \(-0.852623\pi\)
0.894717 0.446633i \(-0.147377\pi\)
\(788\) 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 \(0.0144724\pi\)
−0.998967 + 0.0454506i \(0.985528\pi\)
\(810\) 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 \(0.251555\pi\)
−0.703645 + 0.710552i \(0.748445\pi\)
\(822\) 0 0
\(823\) − 1015.00i − 1.23329i −0.787240 0.616646i \(-0.788491\pi\)
0.787240 0.616646i \(-0.211509\pi\)
\(824\) − 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 \(-0.871234\pi\)
0.919288 0.393586i \(-0.128766\pi\)
\(840\) 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 \(-0.944301\pi\)
0.984729 0.174091i \(-0.0556988\pi\)
\(854\) 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 \(0.365176\pi\)
−0.411011 + 0.911631i \(0.634824\pi\)
\(878\) 236.174 0.268991
\(879\) 0 0
\(880\) 0 0
\(881\) − 961.665i − 1.09156i −0.837928 0.545780i \(-0.816233\pi\)
0.837928 0.545780i \(-0.183767\pi\)
\(882\) 0 0
\(883\) − 193.000i − 0.218573i −0.994010 0.109287i \(-0.965143\pi\)
0.994010 0.109287i \(-0.0348566\pi\)
\(884\) − 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 \(0.0422376\pi\)
−0.991209 + 0.132304i \(0.957762\pi\)
\(908\) −260.215 −0.286581
\(909\) 0 0
\(910\) 0 0
\(911\) 171.120i 0.187837i 0.995580 + 0.0939187i \(0.0299394\pi\)
−0.995580 + 0.0939187i \(0.970061\pi\)
\(912\) 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 \(-0.873365\pi\)
0.921901 0.387424i \(-0.126635\pi\)
\(930\) 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 \(-0.894645\pi\)
0.945723 0.324973i \(-0.105355\pi\)
\(938\) 801.859 0.854860
\(939\) 0 0
\(940\) 0 0
\(941\) − 111.723i − 0.118728i −0.998236 0.0593639i \(-0.981093\pi\)
0.998236 0.0593639i \(-0.0189072\pi\)
\(942\) 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 \(-0.902357\pi\)
0.953319 0.301965i \(-0.0976425\pi\)
\(968\) 1841.31 1.90218
\(969\) 0 0
\(970\) 0 0
\(971\) − 1030.96i − 1.06175i −0.847449 0.530876i \(-0.821863\pi\)
0.847449 0.530876i \(-0.178137\pi\)
\(972\) 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 \(0.0972879\pi\)
−0.953655 + 0.300903i \(0.902712\pi\)
\(998\) −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.224.4 4
3.2 odd 2 inner 225.3.d.a.224.2 4
4.3 odd 2 3600.3.c.e.449.1 4
5.2 odd 4 225.3.c.a.26.2 yes 2
5.3 odd 4 225.3.c.b.26.1 yes 2
5.4 even 2 inner 225.3.d.a.224.1 4
12.11 even 2 3600.3.c.e.449.2 4
15.2 even 4 225.3.c.a.26.1 2
15.8 even 4 225.3.c.b.26.2 yes 2
15.14 odd 2 inner 225.3.d.a.224.3 4
20.3 even 4 3600.3.l.b.1601.1 2
20.7 even 4 3600.3.l.j.1601.1 2
20.19 odd 2 3600.3.c.e.449.3 4
60.23 odd 4 3600.3.l.b.1601.2 2
60.47 odd 4 3600.3.l.j.1601.2 2
60.59 even 2 3600.3.c.e.449.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.3.c.a.26.1 2 15.2 even 4
225.3.c.a.26.2 yes 2 5.2 odd 4
225.3.c.b.26.1 yes 2 5.3 odd 4
225.3.c.b.26.2 yes 2 15.8 even 4
225.3.d.a.224.1 4 5.4 even 2 inner
225.3.d.a.224.2 4 3.2 odd 2 inner
225.3.d.a.224.3 4 15.14 odd 2 inner
225.3.d.a.224.4 4 1.1 even 1 trivial
3600.3.c.e.449.1 4 4.3 odd 2
3600.3.c.e.449.2 4 12.11 even 2
3600.3.c.e.449.3 4 20.19 odd 2
3600.3.c.e.449.4 4 60.59 even 2
3600.3.l.b.1601.1 2 20.3 even 4
3600.3.l.b.1601.2 2 60.23 odd 4
3600.3.l.j.1601.1 2 20.7 even 4
3600.3.l.j.1601.2 2 60.47 odd 4