Properties

Label 1425.2.c.h.799.1
Level $1425$
Weight $2$
Character 1425.799
Analytic conductor $11.379$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1425,2,Mod(799,1425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1425, 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("1425.799");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.c (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: \(11.3786822880\)
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 799.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1425.799
Dual form 1425.2.c.h.799.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{8} -1.00000 q^{9} +5.00000 q^{11} +1.00000i q^{12} -4.00000i q^{13} -1.00000 q^{16} -4.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} -5.00000i q^{22} +9.00000i q^{23} +3.00000 q^{24} -4.00000 q^{26} -1.00000i q^{27} -7.00000 q^{29} +3.00000 q^{31} -5.00000i q^{32} +5.00000i q^{33} -4.00000 q^{34} -1.00000 q^{36} -10.0000i q^{37} -1.00000i q^{38} +4.00000 q^{39} -2.00000 q^{41} -4.00000i q^{43} +5.00000 q^{44} +9.00000 q^{46} +8.00000i q^{47} -1.00000i q^{48} +7.00000 q^{49} +4.00000 q^{51} -4.00000i q^{52} -11.0000i q^{53} -1.00000 q^{54} +1.00000i q^{57} +7.00000i q^{58} -8.00000 q^{59} +13.0000 q^{61} -3.00000i q^{62} -7.00000 q^{64} +5.00000 q^{66} +9.00000i q^{67} -4.00000i q^{68} -9.00000 q^{69} +10.0000 q^{71} +3.00000i q^{72} +5.00000i q^{73} -10.0000 q^{74} +1.00000 q^{76} -4.00000i q^{78} +15.0000 q^{79} +1.00000 q^{81} +2.00000i q^{82} -9.00000i q^{83} -4.00000 q^{86} -7.00000i q^{87} -15.0000i q^{88} -3.00000 q^{89} +9.00000i q^{92} +3.00000i q^{93} +8.00000 q^{94} +5.00000 q^{96} -10.0000i q^{97} -7.00000i q^{98} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9} + 10 q^{11} - 2 q^{16} + 2 q^{19} + 6 q^{24} - 8 q^{26} - 14 q^{29} + 6 q^{31} - 8 q^{34} - 2 q^{36} + 8 q^{39} - 4 q^{41} + 10 q^{44} + 18 q^{46} + 14 q^{49} + 8 q^{51} - 2 q^{54} - 16 q^{59} + 26 q^{61} - 14 q^{64} + 10 q^{66} - 18 q^{69} + 20 q^{71} - 20 q^{74} + 2 q^{76} + 30 q^{79} + 2 q^{81} - 8 q^{86} - 6 q^{89} + 16 q^{94} + 10 q^{96} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times\).

\(n\) \(476\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 −0.935414 0.353553i \(-0.884973\pi\)
0.935414 0.353553i \(-0.115027\pi\)
\(3\) 1.00000i 0.577350i
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) − 3.00000i − 1.06066i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 4.00000i − 1.10940i −0.832050 0.554700i \(-0.812833\pi\)
0.832050 0.554700i \(-0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) − 4.00000i − 0.970143i −0.874475 0.485071i \(-0.838794\pi\)
0.874475 0.485071i \(-0.161206\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) − 5.00000i − 1.06600i
\(23\) 9.00000i 1.87663i 0.345782 + 0.938315i \(0.387614\pi\)
−0.345782 + 0.938315i \(0.612386\pi\)
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) −7.00000 −1.29987 −0.649934 0.759991i \(-0.725203\pi\)
−0.649934 + 0.759991i \(0.725203\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) − 5.00000i − 0.883883i
\(33\) 5.00000i 0.870388i
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) − 10.0000i − 1.64399i −0.569495 0.821995i \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(38\) − 1.00000i − 0.162221i
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 5.00000 0.753778
\(45\) 0 0
\(46\) 9.00000 1.32698
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) − 4.00000i − 0.554700i
\(53\) − 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
\(56\) 0 0
\(57\) 1.00000i 0.132453i
\(58\) 7.00000i 0.919145i
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 13.0000 1.66448 0.832240 0.554416i \(-0.187058\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) − 3.00000i − 0.381000i
\(63\) 0 0
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) 5.00000 0.615457
\(67\) 9.00000i 1.09952i 0.835321 + 0.549762i \(0.185282\pi\)
−0.835321 + 0.549762i \(0.814718\pi\)
\(68\) − 4.00000i − 0.485071i
\(69\) −9.00000 −1.08347
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 3.00000i 0.353553i
\(73\) 5.00000i 0.585206i 0.956234 + 0.292603i \(0.0945214\pi\)
−0.956234 + 0.292603i \(0.905479\pi\)
\(74\) −10.0000 −1.16248
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) − 4.00000i − 0.452911i
\(79\) 15.0000 1.68763 0.843816 0.536633i \(-0.180304\pi\)
0.843816 + 0.536633i \(0.180304\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000i 0.220863i
\(83\) − 9.00000i − 0.987878i −0.869496 0.493939i \(-0.835557\pi\)
0.869496 0.493939i \(-0.164443\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) − 7.00000i − 0.750479i
\(88\) − 15.0000i − 1.59901i
\(89\) −3.00000 −0.317999 −0.159000 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 9.00000i 0.938315i
\(93\) 3.00000i 0.311086i
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 5.00000 0.510310
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) − 7.00000i − 0.707107i
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) − 4.00000i − 0.396059i
\(103\) 13.0000i 1.28093i 0.767988 + 0.640464i \(0.221258\pi\)
−0.767988 + 0.640464i \(0.778742\pi\)
\(104\) −12.0000 −1.17670
\(105\) 0 0
\(106\) −11.0000 −1.06841
\(107\) 6.00000i 0.580042i 0.957020 + 0.290021i \(0.0936623\pi\)
−0.957020 + 0.290021i \(0.906338\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 10.0000 0.949158
\(112\) 0 0
\(113\) 1.00000i 0.0940721i 0.998893 + 0.0470360i \(0.0149776\pi\)
−0.998893 + 0.0470360i \(0.985022\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) −7.00000 −0.649934
\(117\) 4.00000i 0.369800i
\(118\) 8.00000i 0.736460i
\(119\) 0 0
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) − 13.0000i − 1.17696i
\(123\) − 2.00000i − 0.180334i
\(124\) 3.00000 0.269408
\(125\) 0 0
\(126\) 0 0
\(127\) 13.0000i 1.15356i 0.816898 + 0.576782i \(0.195692\pi\)
−0.816898 + 0.576782i \(0.804308\pi\)
\(128\) − 3.00000i − 0.265165i
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 3.00000 0.262111 0.131056 0.991375i \(-0.458163\pi\)
0.131056 + 0.991375i \(0.458163\pi\)
\(132\) 5.00000i 0.435194i
\(133\) 0 0
\(134\) 9.00000 0.777482
\(135\) 0 0
\(136\) −12.0000 −1.02899
\(137\) 12.0000i 1.02523i 0.858619 + 0.512615i \(0.171323\pi\)
−0.858619 + 0.512615i \(0.828677\pi\)
\(138\) 9.00000i 0.766131i
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) − 10.0000i − 0.839181i
\(143\) − 20.0000i − 1.67248i
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 5.00000 0.413803
\(147\) 7.00000i 0.577350i
\(148\) − 10.0000i − 0.821995i
\(149\) −16.0000 −1.31077 −0.655386 0.755295i \(-0.727494\pi\)
−0.655386 + 0.755295i \(0.727494\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) − 3.00000i − 0.243332i
\(153\) 4.00000i 0.323381i
\(154\) 0 0
\(155\) 0 0
\(156\) 4.00000 0.320256
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) − 15.0000i − 1.19334i
\(159\) 11.0000 0.872357
\(160\) 0 0
\(161\) 0 0
\(162\) − 1.00000i − 0.0785674i
\(163\) − 14.0000i − 1.09656i −0.836293 0.548282i \(-0.815282\pi\)
0.836293 0.548282i \(-0.184718\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) −9.00000 −0.698535
\(167\) 6.00000i 0.464294i 0.972681 + 0.232147i \(0.0745750\pi\)
−0.972681 + 0.232147i \(0.925425\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) − 4.00000i − 0.304997i
\(173\) 3.00000i 0.228086i 0.993476 + 0.114043i \(0.0363801\pi\)
−0.993476 + 0.114043i \(0.963620\pi\)
\(174\) −7.00000 −0.530669
\(175\) 0 0
\(176\) −5.00000 −0.376889
\(177\) − 8.00000i − 0.601317i
\(178\) 3.00000i 0.224860i
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 0 0
\(183\) 13.0000i 0.960988i
\(184\) 27.0000 1.99047
\(185\) 0 0
\(186\) 3.00000 0.219971
\(187\) − 20.0000i − 1.46254i
\(188\) 8.00000i 0.583460i
\(189\) 0 0
\(190\) 0 0
\(191\) −17.0000 −1.23008 −0.615038 0.788497i \(-0.710860\pi\)
−0.615038 + 0.788497i \(0.710860\pi\)
\(192\) − 7.00000i − 0.505181i
\(193\) − 14.0000i − 1.00774i −0.863779 0.503871i \(-0.831909\pi\)
0.863779 0.503871i \(-0.168091\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 7.00000 0.500000
\(197\) 12.0000i 0.854965i 0.904024 + 0.427482i \(0.140599\pi\)
−0.904024 + 0.427482i \(0.859401\pi\)
\(198\) 5.00000i 0.355335i
\(199\) −22.0000 −1.55954 −0.779769 0.626067i \(-0.784664\pi\)
−0.779769 + 0.626067i \(0.784664\pi\)
\(200\) 0 0
\(201\) −9.00000 −0.634811
\(202\) 0 0
\(203\) 0 0
\(204\) 4.00000 0.280056
\(205\) 0 0
\(206\) 13.0000 0.905753
\(207\) − 9.00000i − 0.625543i
\(208\) 4.00000i 0.277350i
\(209\) 5.00000 0.345857
\(210\) 0 0
\(211\) 11.0000 0.757271 0.378636 0.925546i \(-0.376393\pi\)
0.378636 + 0.925546i \(0.376393\pi\)
\(212\) − 11.0000i − 0.755483i
\(213\) 10.0000i 0.685189i
\(214\) 6.00000 0.410152
\(215\) 0 0
\(216\) −3.00000 −0.204124
\(217\) 0 0
\(218\) − 10.0000i − 0.677285i
\(219\) −5.00000 −0.337869
\(220\) 0 0
\(221\) −16.0000 −1.07628
\(222\) − 10.0000i − 0.671156i
\(223\) 1.00000i 0.0669650i 0.999439 + 0.0334825i \(0.0106598\pi\)
−0.999439 + 0.0334825i \(0.989340\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1.00000 0.0665190
\(227\) 22.0000i 1.46019i 0.683345 + 0.730096i \(0.260525\pi\)
−0.683345 + 0.730096i \(0.739475\pi\)
\(228\) 1.00000i 0.0662266i
\(229\) −21.0000 −1.38772 −0.693860 0.720110i \(-0.744091\pi\)
−0.693860 + 0.720110i \(0.744091\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 21.0000i 1.37872i
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) −8.00000 −0.520756
\(237\) 15.0000i 0.974355i
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) 4.00000 0.257663 0.128831 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(242\) − 14.0000i − 0.899954i
\(243\) 1.00000i 0.0641500i
\(244\) 13.0000 0.832240
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) − 4.00000i − 0.254514i
\(248\) − 9.00000i − 0.571501i
\(249\) 9.00000 0.570352
\(250\) 0 0
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) 45.0000i 2.82913i
\(254\) 13.0000 0.815693
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 21.0000i 1.30994i 0.755653 + 0.654972i \(0.227320\pi\)
−0.755653 + 0.654972i \(0.772680\pi\)
\(258\) − 4.00000i − 0.249029i
\(259\) 0 0
\(260\) 0 0
\(261\) 7.00000 0.433289
\(262\) − 3.00000i − 0.185341i
\(263\) 17.0000i 1.04826i 0.851637 + 0.524132i \(0.175610\pi\)
−0.851637 + 0.524132i \(0.824390\pi\)
\(264\) 15.0000 0.923186
\(265\) 0 0
\(266\) 0 0
\(267\) − 3.00000i − 0.183597i
\(268\) 9.00000i 0.549762i
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) 10.0000 0.607457 0.303728 0.952759i \(-0.401768\pi\)
0.303728 + 0.952759i \(0.401768\pi\)
\(272\) 4.00000i 0.242536i
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) −9.00000 −0.541736
\(277\) 33.0000i 1.98278i 0.130950 + 0.991389i \(0.458197\pi\)
−0.130950 + 0.991389i \(0.541803\pi\)
\(278\) 14.0000i 0.839664i
\(279\) −3.00000 −0.179605
\(280\) 0 0
\(281\) −15.0000 −0.894825 −0.447412 0.894328i \(-0.647654\pi\)
−0.447412 + 0.894328i \(0.647654\pi\)
\(282\) 8.00000i 0.476393i
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 10.0000 0.593391
\(285\) 0 0
\(286\) −20.0000 −1.18262
\(287\) 0 0
\(288\) 5.00000i 0.294628i
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) 5.00000i 0.292603i
\(293\) − 9.00000i − 0.525786i −0.964825 0.262893i \(-0.915323\pi\)
0.964825 0.262893i \(-0.0846766\pi\)
\(294\) 7.00000 0.408248
\(295\) 0 0
\(296\) −30.0000 −1.74371
\(297\) − 5.00000i − 0.290129i
\(298\) 16.0000i 0.926855i
\(299\) 36.0000 2.08193
\(300\) 0 0
\(301\) 0 0
\(302\) 8.00000i 0.460348i
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 4.00000 0.228665
\(307\) − 23.0000i − 1.31268i −0.754466 0.656340i \(-0.772104\pi\)
0.754466 0.656340i \(-0.227896\pi\)
\(308\) 0 0
\(309\) −13.0000 −0.739544
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) − 12.0000i − 0.679366i
\(313\) 13.0000i 0.734803i 0.930062 + 0.367402i \(0.119753\pi\)
−0.930062 + 0.367402i \(0.880247\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) 15.0000 0.843816
\(317\) 1.00000i 0.0561656i 0.999606 + 0.0280828i \(0.00894021\pi\)
−0.999606 + 0.0280828i \(0.991060\pi\)
\(318\) − 11.0000i − 0.616849i
\(319\) −35.0000 −1.95962
\(320\) 0 0
\(321\) −6.00000 −0.334887
\(322\) 0 0
\(323\) − 4.00000i − 0.222566i
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −14.0000 −0.775388
\(327\) 10.0000i 0.553001i
\(328\) 6.00000i 0.331295i
\(329\) 0 0
\(330\) 0 0
\(331\) −3.00000 −0.164895 −0.0824475 0.996595i \(-0.526274\pi\)
−0.0824475 + 0.996595i \(0.526274\pi\)
\(332\) − 9.00000i − 0.493939i
\(333\) 10.0000i 0.547997i
\(334\) 6.00000 0.328305
\(335\) 0 0
\(336\) 0 0
\(337\) − 8.00000i − 0.435788i −0.975972 0.217894i \(-0.930081\pi\)
0.975972 0.217894i \(-0.0699187\pi\)
\(338\) 3.00000i 0.163178i
\(339\) −1.00000 −0.0543125
\(340\) 0 0
\(341\) 15.0000 0.812296
\(342\) 1.00000i 0.0540738i
\(343\) 0 0
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) 3.00000 0.161281
\(347\) 20.0000i 1.07366i 0.843692 + 0.536828i \(0.180378\pi\)
−0.843692 + 0.536828i \(0.819622\pi\)
\(348\) − 7.00000i − 0.375239i
\(349\) 17.0000 0.909989 0.454995 0.890494i \(-0.349641\pi\)
0.454995 + 0.890494i \(0.349641\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) − 25.0000i − 1.33250i
\(353\) − 12.0000i − 0.638696i −0.947638 0.319348i \(-0.896536\pi\)
0.947638 0.319348i \(-0.103464\pi\)
\(354\) −8.00000 −0.425195
\(355\) 0 0
\(356\) −3.00000 −0.159000
\(357\) 0 0
\(358\) − 4.00000i − 0.211407i
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 16.0000i 0.840941i
\(363\) 14.0000i 0.734809i
\(364\) 0 0
\(365\) 0 0
\(366\) 13.0000 0.679521
\(367\) 18.0000i 0.939592i 0.882775 + 0.469796i \(0.155673\pi\)
−0.882775 + 0.469796i \(0.844327\pi\)
\(368\) − 9.00000i − 0.469157i
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) 0 0
\(372\) 3.00000i 0.155543i
\(373\) 20.0000i 1.03556i 0.855514 + 0.517780i \(0.173242\pi\)
−0.855514 + 0.517780i \(0.826758\pi\)
\(374\) −20.0000 −1.03418
\(375\) 0 0
\(376\) 24.0000 1.23771
\(377\) 28.0000i 1.44207i
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) −13.0000 −0.666010
\(382\) 17.0000i 0.869796i
\(383\) 18.0000i 0.919757i 0.887982 + 0.459879i \(0.152107\pi\)
−0.887982 + 0.459879i \(0.847893\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 4.00000i 0.203331i
\(388\) − 10.0000i − 0.507673i
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) 36.0000 1.82060
\(392\) − 21.0000i − 1.06066i
\(393\) 3.00000i 0.151330i
\(394\) 12.0000 0.604551
\(395\) 0 0
\(396\) −5.00000 −0.251259
\(397\) − 29.0000i − 1.45547i −0.685859 0.727734i \(-0.740573\pi\)
0.685859 0.727734i \(-0.259427\pi\)
\(398\) 22.0000i 1.10276i
\(399\) 0 0
\(400\) 0 0
\(401\) 3.00000 0.149813 0.0749064 0.997191i \(-0.476134\pi\)
0.0749064 + 0.997191i \(0.476134\pi\)
\(402\) 9.00000i 0.448879i
\(403\) − 12.0000i − 0.597763i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 50.0000i − 2.47841i
\(408\) − 12.0000i − 0.594089i
\(409\) 34.0000 1.68119 0.840596 0.541663i \(-0.182205\pi\)
0.840596 + 0.541663i \(0.182205\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) 13.0000i 0.640464i
\(413\) 0 0
\(414\) −9.00000 −0.442326
\(415\) 0 0
\(416\) −20.0000 −0.980581
\(417\) − 14.0000i − 0.685583i
\(418\) − 5.00000i − 0.244558i
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) − 11.0000i − 0.535472i
\(423\) − 8.00000i − 0.388973i
\(424\) −33.0000 −1.60262
\(425\) 0 0
\(426\) 10.0000 0.484502
\(427\) 0 0
\(428\) 6.00000i 0.290021i
\(429\) 20.0000 0.965609
\(430\) 0 0
\(431\) 6.00000 0.289010 0.144505 0.989504i \(-0.453841\pi\)
0.144505 + 0.989504i \(0.453841\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 26.0000i 1.24948i 0.780833 + 0.624740i \(0.214795\pi\)
−0.780833 + 0.624740i \(0.785205\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) 9.00000i 0.430528i
\(438\) 5.00000i 0.238909i
\(439\) −37.0000 −1.76591 −0.882957 0.469454i \(-0.844451\pi\)
−0.882957 + 0.469454i \(0.844451\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 16.0000i 0.761042i
\(443\) 15.0000i 0.712672i 0.934358 + 0.356336i \(0.115974\pi\)
−0.934358 + 0.356336i \(0.884026\pi\)
\(444\) 10.0000 0.474579
\(445\) 0 0
\(446\) 1.00000 0.0473514
\(447\) − 16.0000i − 0.756774i
\(448\) 0 0
\(449\) 7.00000 0.330350 0.165175 0.986264i \(-0.447181\pi\)
0.165175 + 0.986264i \(0.447181\pi\)
\(450\) 0 0
\(451\) −10.0000 −0.470882
\(452\) 1.00000i 0.0470360i
\(453\) − 8.00000i − 0.375873i
\(454\) 22.0000 1.03251
\(455\) 0 0
\(456\) 3.00000 0.140488
\(457\) − 14.0000i − 0.654892i −0.944870 0.327446i \(-0.893812\pi\)
0.944870 0.327446i \(-0.106188\pi\)
\(458\) 21.0000i 0.981266i
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 0 0
\(463\) 12.0000i 0.557687i 0.960337 + 0.278844i \(0.0899511\pi\)
−0.960337 + 0.278844i \(0.910049\pi\)
\(464\) 7.00000 0.324967
\(465\) 0 0
\(466\) 0 0
\(467\) 7.00000i 0.323921i 0.986797 + 0.161961i \(0.0517818\pi\)
−0.986797 + 0.161961i \(0.948218\pi\)
\(468\) 4.00000i 0.184900i
\(469\) 0 0
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) 24.0000i 1.10469i
\(473\) − 20.0000i − 0.919601i
\(474\) 15.0000 0.688973
\(475\) 0 0
\(476\) 0 0
\(477\) 11.0000i 0.503655i
\(478\) 8.00000i 0.365911i
\(479\) −15.0000 −0.685367 −0.342684 0.939451i \(-0.611336\pi\)
−0.342684 + 0.939451i \(0.611336\pi\)
\(480\) 0 0
\(481\) −40.0000 −1.82384
\(482\) − 4.00000i − 0.182195i
\(483\) 0 0
\(484\) 14.0000 0.636364
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 8.00000i 0.362515i 0.983436 + 0.181257i \(0.0580167\pi\)
−0.983436 + 0.181257i \(0.941983\pi\)
\(488\) − 39.0000i − 1.76545i
\(489\) 14.0000 0.633102
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) − 2.00000i − 0.0901670i
\(493\) 28.0000i 1.26106i
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) −3.00000 −0.134704
\(497\) 0 0
\(498\) − 9.00000i − 0.403300i
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) 0 0
\(501\) −6.00000 −0.268060
\(502\) 24.0000i 1.07117i
\(503\) − 28.0000i − 1.24846i −0.781241 0.624229i \(-0.785413\pi\)
0.781241 0.624229i \(-0.214587\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 45.0000 2.00049
\(507\) − 3.00000i − 0.133235i
\(508\) 13.0000i 0.576782i
\(509\) 7.00000 0.310270 0.155135 0.987893i \(-0.450419\pi\)
0.155135 + 0.987893i \(0.450419\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 11.0000i 0.486136i
\(513\) − 1.00000i − 0.0441511i
\(514\) 21.0000 0.926270
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 40.0000i 1.75920i
\(518\) 0 0
\(519\) −3.00000 −0.131685
\(520\) 0 0
\(521\) −41.0000 −1.79624 −0.898121 0.439748i \(-0.855068\pi\)
−0.898121 + 0.439748i \(0.855068\pi\)
\(522\) − 7.00000i − 0.306382i
\(523\) − 28.0000i − 1.22435i −0.790721 0.612177i \(-0.790294\pi\)
0.790721 0.612177i \(-0.209706\pi\)
\(524\) 3.00000 0.131056
\(525\) 0 0
\(526\) 17.0000 0.741235
\(527\) − 12.0000i − 0.522728i
\(528\) − 5.00000i − 0.217597i
\(529\) −58.0000 −2.52174
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) 8.00000i 0.346518i
\(534\) −3.00000 −0.129823
\(535\) 0 0
\(536\) 27.0000 1.16622
\(537\) 4.00000i 0.172613i
\(538\) 14.0000i 0.603583i
\(539\) 35.0000 1.50756
\(540\) 0 0
\(541\) −21.0000 −0.902861 −0.451430 0.892306i \(-0.649086\pi\)
−0.451430 + 0.892306i \(0.649086\pi\)
\(542\) − 10.0000i − 0.429537i
\(543\) − 16.0000i − 0.686626i
\(544\) −20.0000 −0.857493
\(545\) 0 0
\(546\) 0 0
\(547\) 31.0000i 1.32546i 0.748857 + 0.662732i \(0.230603\pi\)
−0.748857 + 0.662732i \(0.769397\pi\)
\(548\) 12.0000i 0.512615i
\(549\) −13.0000 −0.554826
\(550\) 0 0
\(551\) −7.00000 −0.298210
\(552\) 27.0000i 1.14920i
\(553\) 0 0
\(554\) 33.0000 1.40204
\(555\) 0 0
\(556\) −14.0000 −0.593732
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 3.00000i 0.127000i
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 20.0000 0.844401
\(562\) 15.0000i 0.632737i
\(563\) − 30.0000i − 1.26435i −0.774826 0.632175i \(-0.782163\pi\)
0.774826 0.632175i \(-0.217837\pi\)
\(564\) −8.00000 −0.336861
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) − 30.0000i − 1.25877i
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 14.0000 0.585882 0.292941 0.956131i \(-0.405366\pi\)
0.292941 + 0.956131i \(0.405366\pi\)
\(572\) − 20.0000i − 0.836242i
\(573\) − 17.0000i − 0.710185i
\(574\) 0 0
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) − 13.0000i − 0.541197i −0.962692 0.270599i \(-0.912778\pi\)
0.962692 0.270599i \(-0.0872216\pi\)
\(578\) − 1.00000i − 0.0415945i
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) 0 0
\(582\) − 10.0000i − 0.414513i
\(583\) − 55.0000i − 2.27787i
\(584\) 15.0000 0.620704
\(585\) 0 0
\(586\) −9.00000 −0.371787
\(587\) − 3.00000i − 0.123823i −0.998082 0.0619116i \(-0.980280\pi\)
0.998082 0.0619116i \(-0.0197197\pi\)
\(588\) 7.00000i 0.288675i
\(589\) 3.00000 0.123613
\(590\) 0 0
\(591\) −12.0000 −0.493614
\(592\) 10.0000i 0.410997i
\(593\) 20.0000i 0.821302i 0.911793 + 0.410651i \(0.134698\pi\)
−0.911793 + 0.410651i \(0.865302\pi\)
\(594\) −5.00000 −0.205152
\(595\) 0 0
\(596\) −16.0000 −0.655386
\(597\) − 22.0000i − 0.900400i
\(598\) − 36.0000i − 1.47215i
\(599\) 44.0000 1.79779 0.898896 0.438163i \(-0.144371\pi\)
0.898896 + 0.438163i \(0.144371\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) − 9.00000i − 0.366508i
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) 0 0
\(607\) 1.00000i 0.0405887i 0.999794 + 0.0202944i \(0.00646034\pi\)
−0.999794 + 0.0202944i \(0.993540\pi\)
\(608\) − 5.00000i − 0.202777i
\(609\) 0 0
\(610\) 0 0
\(611\) 32.0000 1.29458
\(612\) 4.00000i 0.161690i
\(613\) 26.0000i 1.05013i 0.851062 + 0.525065i \(0.175959\pi\)
−0.851062 + 0.525065i \(0.824041\pi\)
\(614\) −23.0000 −0.928204
\(615\) 0 0
\(616\) 0 0
\(617\) − 42.0000i − 1.69086i −0.534089 0.845428i \(-0.679345\pi\)
0.534089 0.845428i \(-0.320655\pi\)
\(618\) 13.0000i 0.522937i
\(619\) −32.0000 −1.28619 −0.643094 0.765787i \(-0.722350\pi\)
−0.643094 + 0.765787i \(0.722350\pi\)
\(620\) 0 0
\(621\) 9.00000 0.361158
\(622\) − 24.0000i − 0.962312i
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) 0 0
\(626\) 13.0000 0.519584
\(627\) 5.00000i 0.199681i
\(628\) 2.00000i 0.0798087i
\(629\) −40.0000 −1.59490
\(630\) 0 0
\(631\) 28.0000 1.11466 0.557331 0.830290i \(-0.311825\pi\)
0.557331 + 0.830290i \(0.311825\pi\)
\(632\) − 45.0000i − 1.79000i
\(633\) 11.0000i 0.437211i
\(634\) 1.00000 0.0397151
\(635\) 0 0
\(636\) 11.0000 0.436178
\(637\) − 28.0000i − 1.10940i
\(638\) 35.0000i 1.38566i
\(639\) −10.0000 −0.395594
\(640\) 0 0
\(641\) −42.0000 −1.65890 −0.829450 0.558581i \(-0.811346\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 6.00000i 0.236801i
\(643\) − 30.0000i − 1.18308i −0.806274 0.591542i \(-0.798519\pi\)
0.806274 0.591542i \(-0.201481\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −4.00000 −0.157378
\(647\) − 31.0000i − 1.21874i −0.792888 0.609368i \(-0.791423\pi\)
0.792888 0.609368i \(-0.208577\pi\)
\(648\) − 3.00000i − 0.117851i
\(649\) −40.0000 −1.57014
\(650\) 0 0
\(651\) 0 0
\(652\) − 14.0000i − 0.548282i
\(653\) − 16.0000i − 0.626128i −0.949732 0.313064i \(-0.898644\pi\)
0.949732 0.313064i \(-0.101356\pi\)
\(654\) 10.0000 0.391031
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) − 5.00000i − 0.195069i
\(658\) 0 0
\(659\) 4.00000 0.155818 0.0779089 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) 0 0
\(661\) 16.0000 0.622328 0.311164 0.950356i \(-0.399281\pi\)
0.311164 + 0.950356i \(0.399281\pi\)
\(662\) 3.00000i 0.116598i
\(663\) − 16.0000i − 0.621389i
\(664\) −27.0000 −1.04780
\(665\) 0 0
\(666\) 10.0000 0.387492
\(667\) − 63.0000i − 2.43937i
\(668\) 6.00000i 0.232147i
\(669\) −1.00000 −0.0386622
\(670\) 0 0
\(671\) 65.0000 2.50930
\(672\) 0 0
\(673\) − 36.0000i − 1.38770i −0.720121 0.693849i \(-0.755914\pi\)
0.720121 0.693849i \(-0.244086\pi\)
\(674\) −8.00000 −0.308148
\(675\) 0 0
\(676\) −3.00000 −0.115385
\(677\) 27.0000i 1.03769i 0.854867 + 0.518847i \(0.173639\pi\)
−0.854867 + 0.518847i \(0.826361\pi\)
\(678\) 1.00000i 0.0384048i
\(679\) 0 0
\(680\) 0 0
\(681\) −22.0000 −0.843042
\(682\) − 15.0000i − 0.574380i
\(683\) 6.00000i 0.229584i 0.993390 + 0.114792i \(0.0366201\pi\)
−0.993390 + 0.114792i \(0.963380\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) 0 0
\(687\) − 21.0000i − 0.801200i
\(688\) 4.00000i 0.152499i
\(689\) −44.0000 −1.67627
\(690\) 0 0
\(691\) 10.0000 0.380418 0.190209 0.981744i \(-0.439083\pi\)
0.190209 + 0.981744i \(0.439083\pi\)
\(692\) 3.00000i 0.114043i
\(693\) 0 0
\(694\) 20.0000 0.759190
\(695\) 0 0
\(696\) −21.0000 −0.796003
\(697\) 8.00000i 0.303022i
\(698\) − 17.0000i − 0.643459i
\(699\) 0 0
\(700\) 0 0
\(701\) −8.00000 −0.302156 −0.151078 0.988522i \(-0.548274\pi\)
−0.151078 + 0.988522i \(0.548274\pi\)
\(702\) 4.00000i 0.150970i
\(703\) − 10.0000i − 0.377157i
\(704\) −35.0000 −1.31911
\(705\) 0 0
\(706\) −12.0000 −0.451626
\(707\) 0 0
\(708\) − 8.00000i − 0.300658i
\(709\) −13.0000 −0.488225 −0.244113 0.969747i \(-0.578497\pi\)
−0.244113 + 0.969747i \(0.578497\pi\)
\(710\) 0 0
\(711\) −15.0000 −0.562544
\(712\) 9.00000i 0.337289i
\(713\) 27.0000i 1.01116i
\(714\) 0 0
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) − 8.00000i − 0.298765i
\(718\) 0 0
\(719\) −15.0000 −0.559406 −0.279703 0.960087i \(-0.590236\pi\)
−0.279703 + 0.960087i \(0.590236\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 1.00000i − 0.0372161i
\(723\) 4.00000i 0.148762i
\(724\) −16.0000 −0.594635
\(725\) 0 0
\(726\) 14.0000 0.519589
\(727\) 2.00000i 0.0741759i 0.999312 + 0.0370879i \(0.0118082\pi\)
−0.999312 + 0.0370879i \(0.988192\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −16.0000 −0.591781
\(732\) 13.0000i 0.480494i
\(733\) 1.00000i 0.0369358i 0.999829 + 0.0184679i \(0.00587886\pi\)
−0.999829 + 0.0184679i \(0.994121\pi\)
\(734\) 18.0000 0.664392
\(735\) 0 0
\(736\) 45.0000 1.65872
\(737\) 45.0000i 1.65760i
\(738\) − 2.00000i − 0.0736210i
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) 0 0
\(743\) 40.0000i 1.46746i 0.679442 + 0.733729i \(0.262222\pi\)
−0.679442 + 0.733729i \(0.737778\pi\)
\(744\) 9.00000 0.329956
\(745\) 0 0
\(746\) 20.0000 0.732252
\(747\) 9.00000i 0.329293i
\(748\) − 20.0000i − 0.731272i
\(749\) 0 0
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) − 8.00000i − 0.291730i
\(753\) − 24.0000i − 0.874609i
\(754\) 28.0000 1.01970
\(755\) 0 0
\(756\) 0 0
\(757\) 7.00000i 0.254419i 0.991876 + 0.127210i \(0.0406021\pi\)
−0.991876 + 0.127210i \(0.959398\pi\)
\(758\) − 8.00000i − 0.290573i
\(759\) −45.0000 −1.63340
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 13.0000i 0.470940i
\(763\) 0 0
\(764\) −17.0000 −0.615038
\(765\) 0 0
\(766\) 18.0000 0.650366
\(767\) 32.0000i 1.15545i
\(768\) − 17.0000i − 0.613435i
\(769\) 37.0000 1.33425 0.667127 0.744944i \(-0.267524\pi\)
0.667127 + 0.744944i \(0.267524\pi\)
\(770\) 0 0
\(771\) −21.0000 −0.756297
\(772\) − 14.0000i − 0.503871i
\(773\) 18.0000i 0.647415i 0.946157 + 0.323708i \(0.104929\pi\)
−0.946157 + 0.323708i \(0.895071\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) −30.0000 −1.07694
\(777\) 0 0
\(778\) − 30.0000i − 1.07555i
\(779\) −2.00000 −0.0716574
\(780\) 0 0
\(781\) 50.0000 1.78914
\(782\) − 36.0000i − 1.28736i
\(783\) 7.00000i 0.250160i
\(784\) −7.00000 −0.250000
\(785\) 0 0
\(786\) 3.00000 0.107006
\(787\) − 29.0000i − 1.03374i −0.856064 0.516869i \(-0.827097\pi\)
0.856064 0.516869i \(-0.172903\pi\)
\(788\) 12.0000i 0.427482i
\(789\) −17.0000 −0.605216
\(790\) 0 0
\(791\) 0 0
\(792\) 15.0000i 0.533002i
\(793\) − 52.0000i − 1.84657i
\(794\) −29.0000 −1.02917
\(795\) 0 0
\(796\) −22.0000 −0.779769
\(797\) − 14.0000i − 0.495905i −0.968772 0.247953i \(-0.920242\pi\)
0.968772 0.247953i \(-0.0797578\pi\)
\(798\) 0 0
\(799\) 32.0000 1.13208
\(800\) 0 0
\(801\) 3.00000 0.106000
\(802\) − 3.00000i − 0.105934i
\(803\) 25.0000i 0.882231i
\(804\) −9.00000 −0.317406
\(805\) 0 0
\(806\) −12.0000 −0.422682
\(807\) − 14.0000i − 0.492823i
\(808\) 0 0
\(809\) 44.0000 1.54696 0.773479 0.633822i \(-0.218515\pi\)
0.773479 + 0.633822i \(0.218515\pi\)
\(810\) 0 0
\(811\) −51.0000 −1.79085 −0.895426 0.445210i \(-0.853129\pi\)
−0.895426 + 0.445210i \(0.853129\pi\)
\(812\) 0 0
\(813\) 10.0000i 0.350715i
\(814\) −50.0000 −1.75250
\(815\) 0 0
\(816\) −4.00000 −0.140028
\(817\) − 4.00000i − 0.139942i
\(818\) − 34.0000i − 1.18878i
\(819\) 0 0
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 12.0000i 0.418548i
\(823\) − 2.00000i − 0.0697156i −0.999392 0.0348578i \(-0.988902\pi\)
0.999392 0.0348578i \(-0.0110978\pi\)
\(824\) 39.0000 1.35863
\(825\) 0 0
\(826\) 0 0
\(827\) 8.00000i 0.278187i 0.990279 + 0.139094i \(0.0444189\pi\)
−0.990279 + 0.139094i \(0.955581\pi\)
\(828\) − 9.00000i − 0.312772i
\(829\) −30.0000 −1.04194 −0.520972 0.853574i \(-0.674430\pi\)
−0.520972 + 0.853574i \(0.674430\pi\)
\(830\) 0 0
\(831\) −33.0000 −1.14476
\(832\) 28.0000i 0.970725i
\(833\) − 28.0000i − 0.970143i
\(834\) −14.0000 −0.484780
\(835\) 0 0
\(836\) 5.00000 0.172929
\(837\) − 3.00000i − 0.103695i
\(838\) 28.0000i 0.967244i
\(839\) 20.0000 0.690477 0.345238 0.938515i \(-0.387798\pi\)
0.345238 + 0.938515i \(0.387798\pi\)
\(840\) 0 0
\(841\) 20.0000 0.689655
\(842\) 26.0000i 0.896019i
\(843\) − 15.0000i − 0.516627i
\(844\) 11.0000 0.378636
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) 0 0
\(848\) 11.0000i 0.377742i
\(849\) 0 0
\(850\) 0 0
\(851\) 90.0000 3.08516
\(852\) 10.0000i 0.342594i
\(853\) 42.0000i 1.43805i 0.694983 + 0.719026i \(0.255412\pi\)
−0.694983 + 0.719026i \(0.744588\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 18.0000 0.615227
\(857\) − 10.0000i − 0.341593i −0.985306 0.170797i \(-0.945366\pi\)
0.985306 0.170797i \(-0.0546341\pi\)
\(858\) − 20.0000i − 0.682789i
\(859\) −22.0000 −0.750630 −0.375315 0.926897i \(-0.622466\pi\)
−0.375315 + 0.926897i \(0.622466\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 6.00000i − 0.204361i
\(863\) 20.0000i 0.680808i 0.940279 + 0.340404i \(0.110564\pi\)
−0.940279 + 0.340404i \(0.889436\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) 26.0000 0.883516
\(867\) 1.00000i 0.0339618i
\(868\) 0 0
\(869\) 75.0000 2.54420
\(870\) 0 0
\(871\) 36.0000 1.21981
\(872\) − 30.0000i − 1.01593i
\(873\) 10.0000i 0.338449i
\(874\) 9.00000 0.304430
\(875\) 0 0
\(876\) −5.00000 −0.168934
\(877\) − 6.00000i − 0.202606i −0.994856 0.101303i \(-0.967699\pi\)
0.994856 0.101303i \(-0.0323011\pi\)
\(878\) 37.0000i 1.24869i
\(879\) 9.00000 0.303562
\(880\) 0 0
\(881\) 32.0000 1.07811 0.539054 0.842271i \(-0.318782\pi\)
0.539054 + 0.842271i \(0.318782\pi\)
\(882\) 7.00000i 0.235702i
\(883\) 34.0000i 1.14419i 0.820187 + 0.572096i \(0.193869\pi\)
−0.820187 + 0.572096i \(0.806131\pi\)
\(884\) −16.0000 −0.538138
\(885\) 0 0
\(886\) 15.0000 0.503935
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) − 30.0000i − 1.00673i
\(889\) 0 0
\(890\) 0 0
\(891\) 5.00000 0.167506
\(892\) 1.00000i 0.0334825i
\(893\) 8.00000i 0.267710i
\(894\) −16.0000 −0.535120
\(895\) 0 0
\(896\) 0 0
\(897\) 36.0000i 1.20201i
\(898\) − 7.00000i − 0.233593i
\(899\) −21.0000 −0.700389
\(900\) 0 0
\(901\) −44.0000 −1.46585
\(902\) 10.0000i 0.332964i
\(903\) 0 0
\(904\) 3.00000 0.0997785
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) − 24.0000i − 0.796907i −0.917189 0.398453i \(-0.869547\pi\)
0.917189 0.398453i \(-0.130453\pi\)
\(908\) 22.0000i 0.730096i
\(909\) 0 0
\(910\) 0 0
\(911\) 14.0000 0.463841 0.231920 0.972735i \(-0.425499\pi\)
0.231920 + 0.972735i \(0.425499\pi\)
\(912\) − 1.00000i − 0.0331133i
\(913\) − 45.0000i − 1.48928i
\(914\) −14.0000 −0.463079
\(915\) 0 0
\(916\) −21.0000 −0.693860
\(917\) 0 0
\(918\) 4.00000i 0.132020i
\(919\) 26.0000 0.857661 0.428830 0.903385i \(-0.358926\pi\)
0.428830 + 0.903385i \(0.358926\pi\)
\(920\) 0 0
\(921\) 23.0000 0.757876
\(922\) − 12.0000i − 0.395199i
\(923\) − 40.0000i − 1.31662i
\(924\) 0 0
\(925\) 0 0
\(926\) 12.0000 0.394344
\(927\) − 13.0000i − 0.426976i
\(928\) 35.0000i 1.14893i
\(929\) −4.00000 −0.131236 −0.0656179 0.997845i \(-0.520902\pi\)
−0.0656179 + 0.997845i \(0.520902\pi\)
\(930\) 0 0
\(931\) 7.00000 0.229416
\(932\) 0 0
\(933\) 24.0000i 0.785725i
\(934\) 7.00000 0.229047
\(935\) 0 0
\(936\) 12.0000 0.392232
\(937\) 42.0000i 1.37208i 0.727564 + 0.686040i \(0.240653\pi\)
−0.727564 + 0.686040i \(0.759347\pi\)
\(938\) 0 0
\(939\) −13.0000 −0.424239
\(940\) 0 0
\(941\) 13.0000 0.423788 0.211894 0.977293i \(-0.432037\pi\)
0.211894 + 0.977293i \(0.432037\pi\)
\(942\) 2.00000i 0.0651635i
\(943\) − 18.0000i − 0.586161i
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) −20.0000 −0.650256
\(947\) − 12.0000i − 0.389948i −0.980808 0.194974i \(-0.937538\pi\)
0.980808 0.194974i \(-0.0624622\pi\)
\(948\) 15.0000i 0.487177i
\(949\) 20.0000 0.649227
\(950\) 0 0
\(951\) −1.00000 −0.0324272
\(952\) 0 0
\(953\) − 17.0000i − 0.550684i −0.961346 0.275342i \(-0.911209\pi\)
0.961346 0.275342i \(-0.0887911\pi\)
\(954\) 11.0000 0.356138
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) − 35.0000i − 1.13139i
\(958\) 15.0000i 0.484628i
\(959\) 0 0
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 40.0000i 1.28965i
\(963\) − 6.00000i − 0.193347i
\(964\) 4.00000 0.128831
\(965\) 0 0
\(966\) 0 0
\(967\) 2.00000i 0.0643157i 0.999483 + 0.0321578i \(0.0102379\pi\)
−0.999483 + 0.0321578i \(0.989762\pi\)
\(968\) − 42.0000i − 1.34993i
\(969\) 4.00000 0.128499
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 0 0
\(974\) 8.00000 0.256337
\(975\) 0 0
\(976\) −13.0000 −0.416120
\(977\) 2.00000i 0.0639857i 0.999488 + 0.0319928i \(0.0101854\pi\)
−0.999488 + 0.0319928i \(0.989815\pi\)
\(978\) − 14.0000i − 0.447671i
\(979\) −15.0000 −0.479402
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) − 28.0000i − 0.893516i
\(983\) 42.0000i 1.33959i 0.742545 + 0.669796i \(0.233618\pi\)
−0.742545 + 0.669796i \(0.766382\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) 28.0000 0.891702
\(987\) 0 0
\(988\) − 4.00000i − 0.127257i
\(989\) 36.0000 1.14473
\(990\) 0 0
\(991\) 5.00000 0.158830 0.0794151 0.996842i \(-0.474695\pi\)
0.0794151 + 0.996842i \(0.474695\pi\)
\(992\) − 15.0000i − 0.476250i
\(993\) − 3.00000i − 0.0952021i
\(994\) 0 0
\(995\) 0 0
\(996\) 9.00000 0.285176
\(997\) 3.00000i 0.0950110i 0.998871 + 0.0475055i \(0.0151272\pi\)
−0.998871 + 0.0475055i \(0.984873\pi\)
\(998\) − 32.0000i − 1.01294i
\(999\) −10.0000 −0.316386
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 1425.2.c.h.799.1 2
5.2 odd 4 1425.2.a.h.1.1 yes 1
5.3 odd 4 1425.2.a.b.1.1 1
5.4 even 2 inner 1425.2.c.h.799.2 2
15.2 even 4 4275.2.a.f.1.1 1
15.8 even 4 4275.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1425.2.a.b.1.1 1 5.3 odd 4
1425.2.a.h.1.1 yes 1 5.2 odd 4
1425.2.c.h.799.1 2 1.1 even 1 trivial
1425.2.c.h.799.2 2 5.4 even 2 inner
4275.2.a.f.1.1 1 15.2 even 4
4275.2.a.l.1.1 1 15.8 even 4