Properties

Label 2550.2.d.e.2449.1
Level $2550$
Weight $2$
Character 2550.2449
Analytic conductor $20.362$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2550,2,Mod(2449,2550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2550, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2550.2449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2550.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: \(20.3618525154\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2449.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2550.2449
Dual form 2550.2.d.e.2449.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} +1.00000i q^{12} -6.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} +1.00000i q^{18} -4.00000 q^{19} +4.00000 q^{21} +2.00000i q^{22} +5.00000i q^{23} +1.00000 q^{24} -6.00000 q^{26} +1.00000i q^{27} -4.00000i q^{28} +10.0000 q^{31} -1.00000i q^{32} +2.00000i q^{33} +1.00000 q^{34} +1.00000 q^{36} -9.00000i q^{37} +4.00000i q^{38} -6.00000 q^{39} +11.0000 q^{41} -4.00000i q^{42} +10.0000i q^{43} +2.00000 q^{44} +5.00000 q^{46} -8.00000i q^{47} -1.00000i q^{48} -9.00000 q^{49} +1.00000 q^{51} +6.00000i q^{52} -11.0000i q^{53} +1.00000 q^{54} -4.00000 q^{56} +4.00000i q^{57} +15.0000 q^{59} -1.00000 q^{61} -10.0000i q^{62} -4.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} +14.0000i q^{67} -1.00000i q^{68} +5.00000 q^{69} +11.0000 q^{71} -1.00000i q^{72} +8.00000i q^{73} -9.00000 q^{74} +4.00000 q^{76} -8.00000i q^{77} +6.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} -11.0000i q^{82} -5.00000i q^{83} -4.00000 q^{84} +10.0000 q^{86} -2.00000i q^{88} +6.00000 q^{89} +24.0000 q^{91} -5.00000i q^{92} -10.0000i q^{93} -8.00000 q^{94} -1.00000 q^{96} -8.00000i q^{97} +9.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 4 q^{11} + 8 q^{14} + 2 q^{16} - 8 q^{19} + 8 q^{21} + 2 q^{24} - 12 q^{26} + 20 q^{31} + 2 q^{34} + 2 q^{36} - 12 q^{39} + 22 q^{41} + 4 q^{44} + 10 q^{46} - 18 q^{49} + 2 q^{51} + 2 q^{54} - 8 q^{56} + 30 q^{59} - 2 q^{61} - 2 q^{64} + 4 q^{66} + 10 q^{69} + 22 q^{71} - 18 q^{74} + 8 q^{76} + 16 q^{79} + 2 q^{81} - 8 q^{84} + 20 q^{86} + 12 q^{89} + 48 q^{91} - 16 q^{94} - 2 q^{96} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2550\mathbb{Z}\right)^\times\).

\(n\) \(751\) \(851\) \(1327\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) − 1.00000i − 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 6.00000i − 1.66410i −0.554700 0.832050i \(-0.687167\pi\)
0.554700 0.832050i \(-0.312833\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.00000i 0.242536i
\(18\) 1.00000i 0.235702i
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 2.00000i 0.426401i
\(23\) 5.00000i 1.04257i 0.853382 + 0.521286i \(0.174548\pi\)
−0.853382 + 0.521286i \(0.825452\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −6.00000 −1.17670
\(27\) 1.00000i 0.192450i
\(28\) − 4.00000i − 0.755929i
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 2.00000i 0.348155i
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 9.00000i − 1.47959i −0.672832 0.739795i \(-0.734922\pi\)
0.672832 0.739795i \(-0.265078\pi\)
\(38\) 4.00000i 0.648886i
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) 11.0000 1.71791 0.858956 0.512050i \(-0.171114\pi\)
0.858956 + 0.512050i \(0.171114\pi\)
\(42\) − 4.00000i − 0.617213i
\(43\) 10.0000i 1.52499i 0.646997 + 0.762493i \(0.276025\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 5.00000 0.737210
\(47\) − 8.00000i − 1.16692i −0.812142 0.583460i \(-0.801699\pi\)
0.812142 0.583460i \(-0.198301\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 1.00000 0.140028
\(52\) 6.00000i 0.832050i
\(53\) − 11.0000i − 1.51097i −0.655168 0.755483i \(-0.727402\pi\)
0.655168 0.755483i \(-0.272598\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) 4.00000i 0.529813i
\(58\) 0 0
\(59\) 15.0000 1.95283 0.976417 0.215894i \(-0.0692665\pi\)
0.976417 + 0.215894i \(0.0692665\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) − 10.0000i − 1.27000i
\(63\) − 4.00000i − 0.503953i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 14.0000i 1.71037i 0.518321 + 0.855186i \(0.326557\pi\)
−0.518321 + 0.855186i \(0.673443\pi\)
\(68\) − 1.00000i − 0.121268i
\(69\) 5.00000 0.601929
\(70\) 0 0
\(71\) 11.0000 1.30546 0.652730 0.757591i \(-0.273624\pi\)
0.652730 + 0.757591i \(0.273624\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 8.00000i 0.936329i 0.883641 + 0.468165i \(0.155085\pi\)
−0.883641 + 0.468165i \(0.844915\pi\)
\(74\) −9.00000 −1.04623
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) − 8.00000i − 0.911685i
\(78\) 6.00000i 0.679366i
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 11.0000i − 1.21475i
\(83\) − 5.00000i − 0.548821i −0.961613 0.274411i \(-0.911517\pi\)
0.961613 0.274411i \(-0.0884828\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) 10.0000 1.07833
\(87\) 0 0
\(88\) − 2.00000i − 0.213201i
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 24.0000 2.51588
\(92\) − 5.00000i − 0.521286i
\(93\) − 10.0000i − 1.03695i
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) − 8.00000i − 0.812277i −0.913812 0.406138i \(-0.866875\pi\)
0.913812 0.406138i \(-0.133125\pi\)
\(98\) 9.00000i 0.909137i
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) − 1.00000i − 0.0990148i
\(103\) 11.0000i 1.08386i 0.840423 + 0.541931i \(0.182307\pi\)
−0.840423 + 0.541931i \(0.817693\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) −11.0000 −1.06841
\(107\) − 4.00000i − 0.386695i −0.981130 0.193347i \(-0.938066\pi\)
0.981130 0.193347i \(-0.0619344\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) −9.00000 −0.854242
\(112\) 4.00000i 0.377964i
\(113\) − 3.00000i − 0.282216i −0.989994 0.141108i \(-0.954933\pi\)
0.989994 0.141108i \(-0.0450665\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) 0 0
\(117\) 6.00000i 0.554700i
\(118\) − 15.0000i − 1.38086i
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 1.00000i 0.0905357i
\(123\) − 11.0000i − 0.991837i
\(124\) −10.0000 −0.898027
\(125\) 0 0
\(126\) −4.00000 −0.356348
\(127\) − 8.00000i − 0.709885i −0.934888 0.354943i \(-0.884500\pi\)
0.934888 0.354943i \(-0.115500\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 10.0000 0.880451
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) − 2.00000i − 0.174078i
\(133\) − 16.0000i − 1.38738i
\(134\) 14.0000 1.20942
\(135\) 0 0
\(136\) −1.00000 −0.0857493
\(137\) 12.0000i 1.02523i 0.858619 + 0.512615i \(0.171323\pi\)
−0.858619 + 0.512615i \(0.828677\pi\)
\(138\) − 5.00000i − 0.425628i
\(139\) −11.0000 −0.933008 −0.466504 0.884519i \(-0.654487\pi\)
−0.466504 + 0.884519i \(0.654487\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) − 11.0000i − 0.923099i
\(143\) 12.0000i 1.00349i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 8.00000 0.662085
\(147\) 9.00000i 0.742307i
\(148\) 9.00000i 0.739795i
\(149\) 1.00000 0.0819232 0.0409616 0.999161i \(-0.486958\pi\)
0.0409616 + 0.999161i \(0.486958\pi\)
\(150\) 0 0
\(151\) −7.00000 −0.569652 −0.284826 0.958579i \(-0.591936\pi\)
−0.284826 + 0.958579i \(0.591936\pi\)
\(152\) − 4.00000i − 0.324443i
\(153\) − 1.00000i − 0.0808452i
\(154\) −8.00000 −0.644658
\(155\) 0 0
\(156\) 6.00000 0.480384
\(157\) 18.0000i 1.43656i 0.695756 + 0.718278i \(0.255069\pi\)
−0.695756 + 0.718278i \(0.744931\pi\)
\(158\) − 8.00000i − 0.636446i
\(159\) −11.0000 −0.872357
\(160\) 0 0
\(161\) −20.0000 −1.57622
\(162\) − 1.00000i − 0.0785674i
\(163\) − 5.00000i − 0.391630i −0.980641 0.195815i \(-0.937265\pi\)
0.980641 0.195815i \(-0.0627352\pi\)
\(164\) −11.0000 −0.858956
\(165\) 0 0
\(166\) −5.00000 −0.388075
\(167\) − 16.0000i − 1.23812i −0.785345 0.619059i \(-0.787514\pi\)
0.785345 0.619059i \(-0.212486\pi\)
\(168\) 4.00000i 0.308607i
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) − 10.0000i − 0.762493i
\(173\) 4.00000i 0.304114i 0.988372 + 0.152057i \(0.0485898\pi\)
−0.988372 + 0.152057i \(0.951410\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) − 15.0000i − 1.12747i
\(178\) − 6.00000i − 0.449719i
\(179\) −23.0000 −1.71910 −0.859550 0.511051i \(-0.829256\pi\)
−0.859550 + 0.511051i \(0.829256\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) − 24.0000i − 1.77900i
\(183\) 1.00000i 0.0739221i
\(184\) −5.00000 −0.368605
\(185\) 0 0
\(186\) −10.0000 −0.733236
\(187\) − 2.00000i − 0.146254i
\(188\) 8.00000i 0.583460i
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) −14.0000 −1.01300 −0.506502 0.862239i \(-0.669062\pi\)
−0.506502 + 0.862239i \(0.669062\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) − 4.00000i − 0.287926i −0.989583 0.143963i \(-0.954015\pi\)
0.989583 0.143963i \(-0.0459847\pi\)
\(194\) −8.00000 −0.574367
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) − 6.00000i − 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) − 2.00000i − 0.142134i
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 0 0
\(201\) 14.0000 0.987484
\(202\) − 18.0000i − 1.26648i
\(203\) 0 0
\(204\) −1.00000 −0.0700140
\(205\) 0 0
\(206\) 11.0000 0.766406
\(207\) − 5.00000i − 0.347524i
\(208\) − 6.00000i − 0.416025i
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 11.0000i 0.755483i
\(213\) − 11.0000i − 0.753708i
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 40.0000i 2.71538i
\(218\) − 14.0000i − 0.948200i
\(219\) 8.00000 0.540590
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 9.00000i 0.604040i
\(223\) − 7.00000i − 0.468755i −0.972146 0.234377i \(-0.924695\pi\)
0.972146 0.234377i \(-0.0753051\pi\)
\(224\) 4.00000 0.267261
\(225\) 0 0
\(226\) −3.00000 −0.199557
\(227\) − 18.0000i − 1.19470i −0.801980 0.597351i \(-0.796220\pi\)
0.801980 0.597351i \(-0.203780\pi\)
\(228\) − 4.00000i − 0.264906i
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) −8.00000 −0.526361
\(232\) 0 0
\(233\) − 1.00000i − 0.0655122i −0.999463 0.0327561i \(-0.989572\pi\)
0.999463 0.0327561i \(-0.0104285\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) −15.0000 −0.976417
\(237\) − 8.00000i − 0.519656i
\(238\) 4.00000i 0.259281i
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 7.00000i 0.449977i
\(243\) − 1.00000i − 0.0641500i
\(244\) 1.00000 0.0640184
\(245\) 0 0
\(246\) −11.0000 −0.701334
\(247\) 24.0000i 1.52708i
\(248\) 10.0000i 0.635001i
\(249\) −5.00000 −0.316862
\(250\) 0 0
\(251\) 28.0000 1.76734 0.883672 0.468106i \(-0.155064\pi\)
0.883672 + 0.468106i \(0.155064\pi\)
\(252\) 4.00000i 0.251976i
\(253\) − 10.0000i − 0.628695i
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) − 10.0000i − 0.622573i
\(259\) 36.0000 2.23693
\(260\) 0 0
\(261\) 0 0
\(262\) − 12.0000i − 0.741362i
\(263\) 8.00000i 0.493301i 0.969104 + 0.246651i \(0.0793300\pi\)
−0.969104 + 0.246651i \(0.920670\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) −16.0000 −0.981023
\(267\) − 6.00000i − 0.367194i
\(268\) − 14.0000i − 0.855186i
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) 19.0000 1.15417 0.577084 0.816685i \(-0.304191\pi\)
0.577084 + 0.816685i \(0.304191\pi\)
\(272\) 1.00000i 0.0606339i
\(273\) − 24.0000i − 1.45255i
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) −5.00000 −0.300965
\(277\) 3.00000i 0.180253i 0.995930 + 0.0901263i \(0.0287271\pi\)
−0.995930 + 0.0901263i \(0.971273\pi\)
\(278\) 11.0000i 0.659736i
\(279\) −10.0000 −0.598684
\(280\) 0 0
\(281\) 14.0000 0.835170 0.417585 0.908638i \(-0.362877\pi\)
0.417585 + 0.908638i \(0.362877\pi\)
\(282\) 8.00000i 0.476393i
\(283\) 3.00000i 0.178331i 0.996017 + 0.0891657i \(0.0284201\pi\)
−0.996017 + 0.0891657i \(0.971580\pi\)
\(284\) −11.0000 −0.652730
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) 44.0000i 2.59724i
\(288\) 1.00000i 0.0589256i
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −8.00000 −0.468968
\(292\) − 8.00000i − 0.468165i
\(293\) 15.0000i 0.876309i 0.898900 + 0.438155i \(0.144368\pi\)
−0.898900 + 0.438155i \(0.855632\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) 9.00000 0.523114
\(297\) − 2.00000i − 0.116052i
\(298\) − 1.00000i − 0.0579284i
\(299\) 30.0000 1.73494
\(300\) 0 0
\(301\) −40.0000 −2.30556
\(302\) 7.00000i 0.402805i
\(303\) − 18.0000i − 1.03407i
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) −1.00000 −0.0571662
\(307\) − 22.0000i − 1.25561i −0.778372 0.627803i \(-0.783954\pi\)
0.778372 0.627803i \(-0.216046\pi\)
\(308\) 8.00000i 0.455842i
\(309\) 11.0000 0.625768
\(310\) 0 0
\(311\) 1.00000 0.0567048 0.0283524 0.999598i \(-0.490974\pi\)
0.0283524 + 0.999598i \(0.490974\pi\)
\(312\) − 6.00000i − 0.339683i
\(313\) 8.00000i 0.452187i 0.974106 + 0.226093i \(0.0725954\pi\)
−0.974106 + 0.226093i \(0.927405\pi\)
\(314\) 18.0000 1.01580
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 11.0000i 0.616849i
\(319\) 0 0
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 20.0000i 1.11456i
\(323\) − 4.00000i − 0.222566i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −5.00000 −0.276924
\(327\) − 14.0000i − 0.774202i
\(328\) 11.0000i 0.607373i
\(329\) 32.0000 1.76422
\(330\) 0 0
\(331\) 6.00000 0.329790 0.164895 0.986311i \(-0.447272\pi\)
0.164895 + 0.986311i \(0.447272\pi\)
\(332\) 5.00000i 0.274411i
\(333\) 9.00000i 0.493197i
\(334\) −16.0000 −0.875481
\(335\) 0 0
\(336\) 4.00000 0.218218
\(337\) − 10.0000i − 0.544735i −0.962193 0.272367i \(-0.912193\pi\)
0.962193 0.272367i \(-0.0878066\pi\)
\(338\) 23.0000i 1.25104i
\(339\) −3.00000 −0.162938
\(340\) 0 0
\(341\) −20.0000 −1.08306
\(342\) − 4.00000i − 0.216295i
\(343\) − 8.00000i − 0.431959i
\(344\) −10.0000 −0.539164
\(345\) 0 0
\(346\) 4.00000 0.215041
\(347\) − 8.00000i − 0.429463i −0.976673 0.214731i \(-0.931112\pi\)
0.976673 0.214731i \(-0.0688876\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) 2.00000i 0.106600i
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) −15.0000 −0.797241
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 4.00000i 0.211702i
\(358\) 23.0000i 1.21559i
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 7.00000i 0.367912i
\(363\) 7.00000i 0.367405i
\(364\) −24.0000 −1.25794
\(365\) 0 0
\(366\) 1.00000 0.0522708
\(367\) 34.0000i 1.77479i 0.461014 + 0.887393i \(0.347486\pi\)
−0.461014 + 0.887393i \(0.652514\pi\)
\(368\) 5.00000i 0.260643i
\(369\) −11.0000 −0.572637
\(370\) 0 0
\(371\) 44.0000 2.28437
\(372\) 10.0000i 0.518476i
\(373\) − 24.0000i − 1.24267i −0.783544 0.621336i \(-0.786590\pi\)
0.783544 0.621336i \(-0.213410\pi\)
\(374\) −2.00000 −0.103418
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) 0 0
\(378\) 4.00000i 0.205738i
\(379\) −29.0000 −1.48963 −0.744815 0.667271i \(-0.767462\pi\)
−0.744815 + 0.667271i \(0.767462\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 14.0000i 0.716302i
\(383\) 16.0000i 0.817562i 0.912633 + 0.408781i \(0.134046\pi\)
−0.912633 + 0.408781i \(0.865954\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −4.00000 −0.203595
\(387\) − 10.0000i − 0.508329i
\(388\) 8.00000i 0.406138i
\(389\) 33.0000 1.67317 0.836583 0.547840i \(-0.184550\pi\)
0.836583 + 0.547840i \(0.184550\pi\)
\(390\) 0 0
\(391\) −5.00000 −0.252861
\(392\) − 9.00000i − 0.454569i
\(393\) − 12.0000i − 0.605320i
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) 7.00000i 0.351320i 0.984451 + 0.175660i \(0.0562059\pi\)
−0.984451 + 0.175660i \(0.943794\pi\)
\(398\) 10.0000i 0.501255i
\(399\) −16.0000 −0.801002
\(400\) 0 0
\(401\) 23.0000 1.14857 0.574283 0.818657i \(-0.305281\pi\)
0.574283 + 0.818657i \(0.305281\pi\)
\(402\) − 14.0000i − 0.698257i
\(403\) − 60.0000i − 2.98881i
\(404\) −18.0000 −0.895533
\(405\) 0 0
\(406\) 0 0
\(407\) 18.0000i 0.892227i
\(408\) 1.00000i 0.0495074i
\(409\) −19.0000 −0.939490 −0.469745 0.882802i \(-0.655654\pi\)
−0.469745 + 0.882802i \(0.655654\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) − 11.0000i − 0.541931i
\(413\) 60.0000i 2.95241i
\(414\) −5.00000 −0.245737
\(415\) 0 0
\(416\) −6.00000 −0.294174
\(417\) 11.0000i 0.538672i
\(418\) − 8.00000i − 0.391293i
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) − 4.00000i − 0.194717i
\(423\) 8.00000i 0.388973i
\(424\) 11.0000 0.534207
\(425\) 0 0
\(426\) −11.0000 −0.532952
\(427\) − 4.00000i − 0.193574i
\(428\) 4.00000i 0.193347i
\(429\) 12.0000 0.579365
\(430\) 0 0
\(431\) −20.0000 −0.963366 −0.481683 0.876346i \(-0.659974\pi\)
−0.481683 + 0.876346i \(0.659974\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 30.0000i 1.44171i 0.693087 + 0.720854i \(0.256250\pi\)
−0.693087 + 0.720854i \(0.743750\pi\)
\(434\) 40.0000 1.92006
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) − 20.0000i − 0.956730i
\(438\) − 8.00000i − 0.382255i
\(439\) 28.0000 1.33637 0.668184 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) − 6.00000i − 0.285391i
\(443\) − 9.00000i − 0.427603i −0.976877 0.213801i \(-0.931415\pi\)
0.976877 0.213801i \(-0.0685846\pi\)
\(444\) 9.00000 0.427121
\(445\) 0 0
\(446\) −7.00000 −0.331460
\(447\) − 1.00000i − 0.0472984i
\(448\) − 4.00000i − 0.188982i
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) −22.0000 −1.03594
\(452\) 3.00000i 0.141108i
\(453\) 7.00000i 0.328889i
\(454\) −18.0000 −0.844782
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) 15.0000i 0.701670i 0.936437 + 0.350835i \(0.114102\pi\)
−0.936437 + 0.350835i \(0.885898\pi\)
\(458\) − 2.00000i − 0.0934539i
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) −15.0000 −0.698620 −0.349310 0.937007i \(-0.613584\pi\)
−0.349310 + 0.937007i \(0.613584\pi\)
\(462\) 8.00000i 0.372194i
\(463\) 19.0000i 0.883005i 0.897260 + 0.441502i \(0.145554\pi\)
−0.897260 + 0.441502i \(0.854446\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −1.00000 −0.0463241
\(467\) − 37.0000i − 1.71216i −0.516847 0.856078i \(-0.672894\pi\)
0.516847 0.856078i \(-0.327106\pi\)
\(468\) − 6.00000i − 0.277350i
\(469\) −56.0000 −2.58584
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) 15.0000i 0.690431i
\(473\) − 20.0000i − 0.919601i
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) 11.0000i 0.503655i
\(478\) 6.00000i 0.274434i
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −54.0000 −2.46219
\(482\) 14.0000i 0.637683i
\(483\) 20.0000i 0.910032i
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) − 6.00000i − 0.271886i −0.990717 0.135943i \(-0.956594\pi\)
0.990717 0.135943i \(-0.0434064\pi\)
\(488\) − 1.00000i − 0.0452679i
\(489\) −5.00000 −0.226108
\(490\) 0 0
\(491\) 33.0000 1.48927 0.744635 0.667472i \(-0.232624\pi\)
0.744635 + 0.667472i \(0.232624\pi\)
\(492\) 11.0000i 0.495918i
\(493\) 0 0
\(494\) 24.0000 1.07981
\(495\) 0 0
\(496\) 10.0000 0.449013
\(497\) 44.0000i 1.97367i
\(498\) 5.00000i 0.224055i
\(499\) 27.0000 1.20869 0.604343 0.796724i \(-0.293436\pi\)
0.604343 + 0.796724i \(0.293436\pi\)
\(500\) 0 0
\(501\) −16.0000 −0.714827
\(502\) − 28.0000i − 1.24970i
\(503\) − 5.00000i − 0.222939i −0.993768 0.111469i \(-0.964444\pi\)
0.993768 0.111469i \(-0.0355557\pi\)
\(504\) 4.00000 0.178174
\(505\) 0 0
\(506\) −10.0000 −0.444554
\(507\) 23.0000i 1.02147i
\(508\) 8.00000i 0.354943i
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) −32.0000 −1.41560
\(512\) − 1.00000i − 0.0441942i
\(513\) − 4.00000i − 0.176604i
\(514\) 0 0
\(515\) 0 0
\(516\) −10.0000 −0.440225
\(517\) 16.0000i 0.703679i
\(518\) − 36.0000i − 1.58175i
\(519\) 4.00000 0.175581
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) − 6.00000i − 0.262362i −0.991358 0.131181i \(-0.958123\pi\)
0.991358 0.131181i \(-0.0418769\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 8.00000 0.348817
\(527\) 10.0000i 0.435607i
\(528\) 2.00000i 0.0870388i
\(529\) −2.00000 −0.0869565
\(530\) 0 0
\(531\) −15.0000 −0.650945
\(532\) 16.0000i 0.693688i
\(533\) − 66.0000i − 2.85878i
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) −14.0000 −0.604708
\(537\) 23.0000i 0.992523i
\(538\) 10.0000i 0.431131i
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) 21.0000 0.902861 0.451430 0.892306i \(-0.350914\pi\)
0.451430 + 0.892306i \(0.350914\pi\)
\(542\) − 19.0000i − 0.816120i
\(543\) 7.00000i 0.300399i
\(544\) 1.00000 0.0428746
\(545\) 0 0
\(546\) −24.0000 −1.02711
\(547\) − 15.0000i − 0.641354i −0.947189 0.320677i \(-0.896090\pi\)
0.947189 0.320677i \(-0.103910\pi\)
\(548\) − 12.0000i − 0.512615i
\(549\) 1.00000 0.0426790
\(550\) 0 0
\(551\) 0 0
\(552\) 5.00000i 0.212814i
\(553\) 32.0000i 1.36078i
\(554\) 3.00000 0.127458
\(555\) 0 0
\(556\) 11.0000 0.466504
\(557\) − 15.0000i − 0.635570i −0.948163 0.317785i \(-0.897061\pi\)
0.948163 0.317785i \(-0.102939\pi\)
\(558\) 10.0000i 0.423334i
\(559\) 60.0000 2.53773
\(560\) 0 0
\(561\) −2.00000 −0.0844401
\(562\) − 14.0000i − 0.590554i
\(563\) 41.0000i 1.72794i 0.503540 + 0.863972i \(0.332031\pi\)
−0.503540 + 0.863972i \(0.667969\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) 3.00000 0.126099
\(567\) 4.00000i 0.167984i
\(568\) 11.0000i 0.461550i
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) −19.0000 −0.795125 −0.397563 0.917575i \(-0.630144\pi\)
−0.397563 + 0.917575i \(0.630144\pi\)
\(572\) − 12.0000i − 0.501745i
\(573\) 14.0000i 0.584858i
\(574\) 44.0000 1.83652
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 31.0000i − 1.29055i −0.763952 0.645273i \(-0.776743\pi\)
0.763952 0.645273i \(-0.223257\pi\)
\(578\) 1.00000i 0.0415945i
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) 20.0000 0.829740
\(582\) 8.00000i 0.331611i
\(583\) 22.0000i 0.911147i
\(584\) −8.00000 −0.331042
\(585\) 0 0
\(586\) 15.0000 0.619644
\(587\) − 23.0000i − 0.949312i −0.880172 0.474656i \(-0.842573\pi\)
0.880172 0.474656i \(-0.157427\pi\)
\(588\) − 9.00000i − 0.371154i
\(589\) −40.0000 −1.64817
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) − 9.00000i − 0.369898i
\(593\) − 16.0000i − 0.657041i −0.944497 0.328521i \(-0.893450\pi\)
0.944497 0.328521i \(-0.106550\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) −1.00000 −0.0409616
\(597\) 10.0000i 0.409273i
\(598\) − 30.0000i − 1.22679i
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 40.0000i 1.63028i
\(603\) − 14.0000i − 0.570124i
\(604\) 7.00000 0.284826
\(605\) 0 0
\(606\) −18.0000 −0.731200
\(607\) 10.0000i 0.405887i 0.979190 + 0.202944i \(0.0650509\pi\)
−0.979190 + 0.202944i \(0.934949\pi\)
\(608\) 4.00000i 0.162221i
\(609\) 0 0
\(610\) 0 0
\(611\) −48.0000 −1.94187
\(612\) 1.00000i 0.0404226i
\(613\) − 42.0000i − 1.69636i −0.529705 0.848182i \(-0.677697\pi\)
0.529705 0.848182i \(-0.322303\pi\)
\(614\) −22.0000 −0.887848
\(615\) 0 0
\(616\) 8.00000 0.322329
\(617\) − 22.0000i − 0.885687i −0.896599 0.442843i \(-0.853970\pi\)
0.896599 0.442843i \(-0.146030\pi\)
\(618\) − 11.0000i − 0.442485i
\(619\) 36.0000 1.44696 0.723481 0.690344i \(-0.242541\pi\)
0.723481 + 0.690344i \(0.242541\pi\)
\(620\) 0 0
\(621\) −5.00000 −0.200643
\(622\) − 1.00000i − 0.0400963i
\(623\) 24.0000i 0.961540i
\(624\) −6.00000 −0.240192
\(625\) 0 0
\(626\) 8.00000 0.319744
\(627\) − 8.00000i − 0.319489i
\(628\) − 18.0000i − 0.718278i
\(629\) 9.00000 0.358854
\(630\) 0 0
\(631\) −15.0000 −0.597141 −0.298570 0.954388i \(-0.596510\pi\)
−0.298570 + 0.954388i \(0.596510\pi\)
\(632\) 8.00000i 0.318223i
\(633\) − 4.00000i − 0.158986i
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) 11.0000 0.436178
\(637\) 54.0000i 2.13956i
\(638\) 0 0
\(639\) −11.0000 −0.435153
\(640\) 0 0
\(641\) 10.0000 0.394976 0.197488 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) 4.00000i 0.157867i
\(643\) 23.0000i 0.907031i 0.891248 + 0.453516i \(0.149830\pi\)
−0.891248 + 0.453516i \(0.850170\pi\)
\(644\) 20.0000 0.788110
\(645\) 0 0
\(646\) −4.00000 −0.157378
\(647\) − 30.0000i − 1.17942i −0.807614 0.589711i \(-0.799242\pi\)
0.807614 0.589711i \(-0.200758\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −30.0000 −1.17760
\(650\) 0 0
\(651\) 40.0000 1.56772
\(652\) 5.00000i 0.195815i
\(653\) 16.0000i 0.626128i 0.949732 + 0.313064i \(0.101356\pi\)
−0.949732 + 0.313064i \(0.898644\pi\)
\(654\) −14.0000 −0.547443
\(655\) 0 0
\(656\) 11.0000 0.429478
\(657\) − 8.00000i − 0.312110i
\(658\) − 32.0000i − 1.24749i
\(659\) 24.0000 0.934907 0.467454 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) 0 0
\(661\) −42.0000 −1.63361 −0.816805 0.576913i \(-0.804257\pi\)
−0.816805 + 0.576913i \(0.804257\pi\)
\(662\) − 6.00000i − 0.233197i
\(663\) − 6.00000i − 0.233021i
\(664\) 5.00000 0.194038
\(665\) 0 0
\(666\) 9.00000 0.348743
\(667\) 0 0
\(668\) 16.0000i 0.619059i
\(669\) −7.00000 −0.270636
\(670\) 0 0
\(671\) 2.00000 0.0772091
\(672\) − 4.00000i − 0.154303i
\(673\) 36.0000i 1.38770i 0.720121 + 0.693849i \(0.244086\pi\)
−0.720121 + 0.693849i \(0.755914\pi\)
\(674\) −10.0000 −0.385186
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) 48.0000i 1.84479i 0.386248 + 0.922395i \(0.373771\pi\)
−0.386248 + 0.922395i \(0.626229\pi\)
\(678\) 3.00000i 0.115214i
\(679\) 32.0000 1.22805
\(680\) 0 0
\(681\) −18.0000 −0.689761
\(682\) 20.0000i 0.765840i
\(683\) 34.0000i 1.30097i 0.759517 + 0.650487i \(0.225435\pi\)
−0.759517 + 0.650487i \(0.774565\pi\)
\(684\) −4.00000 −0.152944
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) − 2.00000i − 0.0763048i
\(688\) 10.0000i 0.381246i
\(689\) −66.0000 −2.51440
\(690\) 0 0
\(691\) −35.0000 −1.33146 −0.665731 0.746191i \(-0.731880\pi\)
−0.665731 + 0.746191i \(0.731880\pi\)
\(692\) − 4.00000i − 0.152057i
\(693\) 8.00000i 0.303895i
\(694\) −8.00000 −0.303676
\(695\) 0 0
\(696\) 0 0
\(697\) 11.0000i 0.416655i
\(698\) − 14.0000i − 0.529908i
\(699\) −1.00000 −0.0378235
\(700\) 0 0
\(701\) −5.00000 −0.188847 −0.0944237 0.995532i \(-0.530101\pi\)
−0.0944237 + 0.995532i \(0.530101\pi\)
\(702\) − 6.00000i − 0.226455i
\(703\) 36.0000i 1.35777i
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) 0 0
\(707\) 72.0000i 2.70784i
\(708\) 15.0000i 0.563735i
\(709\) −2.00000 −0.0751116 −0.0375558 0.999295i \(-0.511957\pi\)
−0.0375558 + 0.999295i \(0.511957\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 6.00000i 0.224860i
\(713\) 50.0000i 1.87251i
\(714\) 4.00000 0.149696
\(715\) 0 0
\(716\) 23.0000 0.859550
\(717\) 6.00000i 0.224074i
\(718\) 8.00000i 0.298557i
\(719\) 40.0000 1.49175 0.745874 0.666087i \(-0.232032\pi\)
0.745874 + 0.666087i \(0.232032\pi\)
\(720\) 0 0
\(721\) −44.0000 −1.63865
\(722\) 3.00000i 0.111648i
\(723\) 14.0000i 0.520666i
\(724\) 7.00000 0.260153
\(725\) 0 0
\(726\) 7.00000 0.259794
\(727\) 28.0000i 1.03846i 0.854634 + 0.519231i \(0.173782\pi\)
−0.854634 + 0.519231i \(0.826218\pi\)
\(728\) 24.0000i 0.889499i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −10.0000 −0.369863
\(732\) − 1.00000i − 0.0369611i
\(733\) 26.0000i 0.960332i 0.877178 + 0.480166i \(0.159424\pi\)
−0.877178 + 0.480166i \(0.840576\pi\)
\(734\) 34.0000 1.25496
\(735\) 0 0
\(736\) 5.00000 0.184302
\(737\) − 28.0000i − 1.03139i
\(738\) 11.0000i 0.404916i
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 24.0000 0.881662
\(742\) − 44.0000i − 1.61529i
\(743\) 37.0000i 1.35740i 0.734416 + 0.678699i \(0.237456\pi\)
−0.734416 + 0.678699i \(0.762544\pi\)
\(744\) 10.0000 0.366618
\(745\) 0 0
\(746\) −24.0000 −0.878702
\(747\) 5.00000i 0.182940i
\(748\) 2.00000i 0.0731272i
\(749\) 16.0000 0.584627
\(750\) 0 0
\(751\) −24.0000 −0.875772 −0.437886 0.899030i \(-0.644273\pi\)
−0.437886 + 0.899030i \(0.644273\pi\)
\(752\) − 8.00000i − 0.291730i
\(753\) − 28.0000i − 1.02038i
\(754\) 0 0
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) − 40.0000i − 1.45382i −0.686730 0.726912i \(-0.740955\pi\)
0.686730 0.726912i \(-0.259045\pi\)
\(758\) 29.0000i 1.05333i
\(759\) −10.0000 −0.362977
\(760\) 0 0
\(761\) 8.00000 0.290000 0.145000 0.989432i \(-0.453682\pi\)
0.145000 + 0.989432i \(0.453682\pi\)
\(762\) 8.00000i 0.289809i
\(763\) 56.0000i 2.02734i
\(764\) 14.0000 0.506502
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) − 90.0000i − 3.24971i
\(768\) − 1.00000i − 0.0360844i
\(769\) 5.00000 0.180305 0.0901523 0.995928i \(-0.471265\pi\)
0.0901523 + 0.995928i \(0.471265\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4.00000i 0.143963i
\(773\) − 17.0000i − 0.611448i −0.952120 0.305724i \(-0.901102\pi\)
0.952120 0.305724i \(-0.0988984\pi\)
\(774\) −10.0000 −0.359443
\(775\) 0 0
\(776\) 8.00000 0.287183
\(777\) − 36.0000i − 1.29149i
\(778\) − 33.0000i − 1.18311i
\(779\) −44.0000 −1.57646
\(780\) 0 0
\(781\) −22.0000 −0.787222
\(782\) 5.00000i 0.178800i
\(783\) 0 0
\(784\) −9.00000 −0.321429
\(785\) 0 0
\(786\) −12.0000 −0.428026
\(787\) − 1.00000i − 0.0356462i −0.999841 0.0178231i \(-0.994326\pi\)
0.999841 0.0178231i \(-0.00567356\pi\)
\(788\) 6.00000i 0.213741i
\(789\) 8.00000 0.284808
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 2.00000i 0.0710669i
\(793\) 6.00000i 0.213066i
\(794\) 7.00000 0.248421
\(795\) 0 0
\(796\) 10.0000 0.354441
\(797\) − 15.0000i − 0.531327i −0.964066 0.265664i \(-0.914409\pi\)
0.964066 0.265664i \(-0.0855911\pi\)
\(798\) 16.0000i 0.566394i
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) − 23.0000i − 0.812158i
\(803\) − 16.0000i − 0.564628i
\(804\) −14.0000 −0.493742
\(805\) 0 0
\(806\) −60.0000 −2.11341
\(807\) 10.0000i 0.352017i
\(808\) 18.0000i 0.633238i
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) −4.00000 −0.140459 −0.0702295 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) 0 0
\(813\) − 19.0000i − 0.666359i
\(814\) 18.0000 0.630900
\(815\) 0 0
\(816\) 1.00000 0.0350070
\(817\) − 40.0000i − 1.39942i
\(818\) 19.0000i 0.664319i
\(819\) −24.0000 −0.838628
\(820\) 0 0
\(821\) −24.0000 −0.837606 −0.418803 0.908077i \(-0.637550\pi\)
−0.418803 + 0.908077i \(0.637550\pi\)
\(822\) − 12.0000i − 0.418548i
\(823\) − 40.0000i − 1.39431i −0.716919 0.697156i \(-0.754448\pi\)
0.716919 0.697156i \(-0.245552\pi\)
\(824\) −11.0000 −0.383203
\(825\) 0 0
\(826\) 60.0000 2.08767
\(827\) − 18.0000i − 0.625921i −0.949766 0.312961i \(-0.898679\pi\)
0.949766 0.312961i \(-0.101321\pi\)
\(828\) 5.00000i 0.173762i
\(829\) −4.00000 −0.138926 −0.0694629 0.997585i \(-0.522129\pi\)
−0.0694629 + 0.997585i \(0.522129\pi\)
\(830\) 0 0
\(831\) 3.00000 0.104069
\(832\) 6.00000i 0.208013i
\(833\) − 9.00000i − 0.311832i
\(834\) 11.0000 0.380899
\(835\) 0 0
\(836\) −8.00000 −0.276686
\(837\) 10.0000i 0.345651i
\(838\) − 4.00000i − 0.138178i
\(839\) −15.0000 −0.517858 −0.258929 0.965896i \(-0.583369\pi\)
−0.258929 + 0.965896i \(0.583369\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) − 8.00000i − 0.275698i
\(843\) − 14.0000i − 0.482186i
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) − 28.0000i − 0.962091i
\(848\) − 11.0000i − 0.377742i
\(849\) 3.00000 0.102960
\(850\) 0 0
\(851\) 45.0000 1.54258
\(852\) 11.0000i 0.376854i
\(853\) − 30.0000i − 1.02718i −0.858036 0.513590i \(-0.828315\pi\)
0.858036 0.513590i \(-0.171685\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) 4.00000 0.136717
\(857\) − 19.0000i − 0.649028i −0.945881 0.324514i \(-0.894799\pi\)
0.945881 0.324514i \(-0.105201\pi\)
\(858\) − 12.0000i − 0.409673i
\(859\) −28.0000 −0.955348 −0.477674 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) 0 0
\(861\) 44.0000 1.49952
\(862\) 20.0000i 0.681203i
\(863\) 28.0000i 0.953131i 0.879139 + 0.476566i \(0.158119\pi\)
−0.879139 + 0.476566i \(0.841881\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 30.0000 1.01944
\(867\) 1.00000i 0.0339618i
\(868\) − 40.0000i − 1.35769i
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) 84.0000 2.84623
\(872\) 14.0000i 0.474100i
\(873\) 8.00000i 0.270759i
\(874\) −20.0000 −0.676510
\(875\) 0 0
\(876\) −8.00000 −0.270295
\(877\) 18.0000i 0.607817i 0.952701 + 0.303908i \(0.0982917\pi\)
−0.952701 + 0.303908i \(0.901708\pi\)
\(878\) − 28.0000i − 0.944954i
\(879\) 15.0000 0.505937
\(880\) 0 0
\(881\) −19.0000 −0.640126 −0.320063 0.947396i \(-0.603704\pi\)
−0.320063 + 0.947396i \(0.603704\pi\)
\(882\) − 9.00000i − 0.303046i
\(883\) − 34.0000i − 1.14419i −0.820187 0.572096i \(-0.806131\pi\)
0.820187 0.572096i \(-0.193869\pi\)
\(884\) −6.00000 −0.201802
\(885\) 0 0
\(886\) −9.00000 −0.302361
\(887\) − 27.0000i − 0.906571i −0.891365 0.453286i \(-0.850252\pi\)
0.891365 0.453286i \(-0.149748\pi\)
\(888\) − 9.00000i − 0.302020i
\(889\) 32.0000 1.07325
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) 7.00000i 0.234377i
\(893\) 32.0000i 1.07084i
\(894\) −1.00000 −0.0334450
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) − 30.0000i − 1.00167i
\(898\) 6.00000i 0.200223i
\(899\) 0 0
\(900\) 0 0
\(901\) 11.0000 0.366463
\(902\) 22.0000i 0.732520i
\(903\) 40.0000i 1.33112i
\(904\) 3.00000 0.0997785
\(905\) 0 0
\(906\) 7.00000 0.232559
\(907\) − 13.0000i − 0.431658i −0.976431 0.215829i \(-0.930755\pi\)
0.976431 0.215829i \(-0.0692454\pi\)
\(908\) 18.0000i 0.597351i
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 4.00000i 0.132453i
\(913\) 10.0000i 0.330952i
\(914\) 15.0000 0.496156
\(915\) 0 0
\(916\) −2.00000 −0.0660819
\(917\) 48.0000i 1.58510i
\(918\) 1.00000i 0.0330049i
\(919\) −1.00000 −0.0329870 −0.0164935 0.999864i \(-0.505250\pi\)
−0.0164935 + 0.999864i \(0.505250\pi\)
\(920\) 0 0
\(921\) −22.0000 −0.724925
\(922\) 15.0000i 0.493999i
\(923\) − 66.0000i − 2.17242i
\(924\) 8.00000 0.263181
\(925\) 0 0
\(926\) 19.0000 0.624379
\(927\) − 11.0000i − 0.361287i
\(928\) 0 0
\(929\) −37.0000 −1.21393 −0.606965 0.794728i \(-0.707613\pi\)
−0.606965 + 0.794728i \(0.707613\pi\)
\(930\) 0 0
\(931\) 36.0000 1.17985
\(932\) 1.00000i 0.0327561i
\(933\) − 1.00000i − 0.0327385i
\(934\) −37.0000 −1.21068
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) − 23.0000i − 0.751377i −0.926746 0.375689i \(-0.877406\pi\)
0.926746 0.375689i \(-0.122594\pi\)
\(938\) 56.0000i 1.82846i
\(939\) 8.00000 0.261070
\(940\) 0 0
\(941\) 42.0000 1.36916 0.684580 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) − 18.0000i − 0.586472i
\(943\) 55.0000i 1.79105i
\(944\) 15.0000 0.488208
\(945\) 0 0
\(946\) −20.0000 −0.650256
\(947\) 24.0000i 0.779895i 0.920837 + 0.389948i \(0.127507\pi\)
−0.920837 + 0.389948i \(0.872493\pi\)
\(948\) 8.00000i 0.259828i
\(949\) 48.0000 1.55815
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) − 4.00000i − 0.129641i
\(953\) − 18.0000i − 0.583077i −0.956559 0.291539i \(-0.905833\pi\)
0.956559 0.291539i \(-0.0941672\pi\)
\(954\) 11.0000 0.356138
\(955\) 0 0
\(956\) 6.00000 0.194054
\(957\) 0 0
\(958\) − 24.0000i − 0.775405i
\(959\) −48.0000 −1.55000
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 54.0000i 1.74103i
\(963\) 4.00000i 0.128898i
\(964\) 14.0000 0.450910
\(965\) 0 0
\(966\) 20.0000 0.643489
\(967\) 31.0000i 0.996893i 0.866921 + 0.498446i \(0.166096\pi\)
−0.866921 + 0.498446i \(0.833904\pi\)
\(968\) − 7.00000i − 0.224989i
\(969\) −4.00000 −0.128499
\(970\) 0 0
\(971\) −49.0000 −1.57248 −0.786242 0.617918i \(-0.787976\pi\)
−0.786242 + 0.617918i \(0.787976\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) − 44.0000i − 1.41058i
\(974\) −6.00000 −0.192252
\(975\) 0 0
\(976\) −1.00000 −0.0320092
\(977\) 42.0000i 1.34370i 0.740688 + 0.671850i \(0.234500\pi\)
−0.740688 + 0.671850i \(0.765500\pi\)
\(978\) 5.00000i 0.159882i
\(979\) −12.0000 −0.383522
\(980\) 0 0
\(981\) −14.0000 −0.446986
\(982\) − 33.0000i − 1.05307i
\(983\) − 24.0000i − 0.765481i −0.923856 0.382741i \(-0.874980\pi\)
0.923856 0.382741i \(-0.125020\pi\)
\(984\) 11.0000 0.350667
\(985\) 0 0
\(986\) 0 0
\(987\) − 32.0000i − 1.01857i
\(988\) − 24.0000i − 0.763542i
\(989\) −50.0000 −1.58991
\(990\) 0 0
\(991\) −60.0000 −1.90596 −0.952981 0.303029i \(-0.902002\pi\)
−0.952981 + 0.303029i \(0.902002\pi\)
\(992\) − 10.0000i − 0.317500i
\(993\) − 6.00000i − 0.190404i
\(994\) 44.0000 1.39560
\(995\) 0 0
\(996\) 5.00000 0.158431
\(997\) 10.0000i 0.316703i 0.987383 + 0.158352i \(0.0506179\pi\)
−0.987383 + 0.158352i \(0.949382\pi\)
\(998\) − 27.0000i − 0.854670i
\(999\) 9.00000 0.284747
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 2550.2.d.e.2449.1 2
5.2 odd 4 2550.2.a.r.1.1 yes 1
5.3 odd 4 2550.2.a.p.1.1 1
5.4 even 2 inner 2550.2.d.e.2449.2 2
15.2 even 4 7650.2.a.d.1.1 1
15.8 even 4 7650.2.a.cm.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2550.2.a.p.1.1 1 5.3 odd 4
2550.2.a.r.1.1 yes 1 5.2 odd 4
2550.2.d.e.2449.1 2 1.1 even 1 trivial
2550.2.d.e.2449.2 2 5.4 even 2 inner
7650.2.a.d.1.1 1 15.2 even 4
7650.2.a.cm.1.1 1 15.8 even 4