Properties

Label 525.2.a.f.1.1
Level $525$
Weight $2$
Character 525.1
Self dual yes
Analytic conductor $4.192$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,2,Mod(1,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.a (trivial)

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: yes
Analytic conductor: \(4.19214610612\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 525.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.30278 q^{2} -1.00000 q^{3} +3.30278 q^{4} +2.30278 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.30278 q^{2} -1.00000 q^{3} +3.30278 q^{4} +2.30278 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -3.00000 q^{11} -3.30278 q^{12} +2.60555 q^{13} +2.30278 q^{14} +0.302776 q^{16} -4.60555 q^{17} -2.30278 q^{18} +6.60555 q^{19} +1.00000 q^{21} +6.90833 q^{22} +6.21110 q^{23} +3.00000 q^{24} -6.00000 q^{26} -1.00000 q^{27} -3.30278 q^{28} -7.60555 q^{29} -7.21110 q^{31} +5.30278 q^{32} +3.00000 q^{33} +10.6056 q^{34} +3.30278 q^{36} +4.21110 q^{37} -15.2111 q^{38} -2.60555 q^{39} -2.30278 q^{42} -9.60555 q^{43} -9.90833 q^{44} -14.3028 q^{46} -10.6056 q^{47} -0.302776 q^{48} +1.00000 q^{49} +4.60555 q^{51} +8.60555 q^{52} -3.21110 q^{53} +2.30278 q^{54} +3.00000 q^{56} -6.60555 q^{57} +17.5139 q^{58} -10.6056 q^{59} -1.21110 q^{61} +16.6056 q^{62} -1.00000 q^{63} -12.8167 q^{64} -6.90833 q^{66} -15.6056 q^{67} -15.2111 q^{68} -6.21110 q^{69} -3.00000 q^{71} -3.00000 q^{72} -0.605551 q^{73} -9.69722 q^{74} +21.8167 q^{76} +3.00000 q^{77} +6.00000 q^{78} -14.8167 q^{79} +1.00000 q^{81} +3.21110 q^{83} +3.30278 q^{84} +22.1194 q^{86} +7.60555 q^{87} +9.00000 q^{88} +7.81665 q^{89} -2.60555 q^{91} +20.5139 q^{92} +7.21110 q^{93} +24.4222 q^{94} -5.30278 q^{96} +0.788897 q^{97} -2.30278 q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 2 q^{3} + 3 q^{4} + q^{6} - 2 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 2 q^{3} + 3 q^{4} + q^{6} - 2 q^{7} - 6 q^{8} + 2 q^{9} - 6 q^{11} - 3 q^{12} - 2 q^{13} + q^{14} - 3 q^{16} - 2 q^{17} - q^{18} + 6 q^{19} + 2 q^{21} + 3 q^{22} - 2 q^{23} + 6 q^{24} - 12 q^{26} - 2 q^{27} - 3 q^{28} - 8 q^{29} + 7 q^{32} + 6 q^{33} + 14 q^{34} + 3 q^{36} - 6 q^{37} - 16 q^{38} + 2 q^{39} - q^{42} - 12 q^{43} - 9 q^{44} - 25 q^{46} - 14 q^{47} + 3 q^{48} + 2 q^{49} + 2 q^{51} + 10 q^{52} + 8 q^{53} + q^{54} + 6 q^{56} - 6 q^{57} + 17 q^{58} - 14 q^{59} + 12 q^{61} + 26 q^{62} - 2 q^{63} - 4 q^{64} - 3 q^{66} - 24 q^{67} - 16 q^{68} + 2 q^{69} - 6 q^{71} - 6 q^{72} + 6 q^{73} - 23 q^{74} + 22 q^{76} + 6 q^{77} + 12 q^{78} - 8 q^{79} + 2 q^{81} - 8 q^{83} + 3 q^{84} + 19 q^{86} + 8 q^{87} + 18 q^{88} - 6 q^{89} + 2 q^{91} + 23 q^{92} + 20 q^{94} - 7 q^{96} + 16 q^{97} - q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.30278 −1.62831 −0.814154 0.580649i \(-0.802799\pi\)
−0.814154 + 0.580649i \(0.802799\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.30278 1.65139
\(5\) 0 0
\(6\) 2.30278 0.940104
\(7\) −1.00000 −0.377964
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) −3.30278 −0.953429
\(13\) 2.60555 0.722650 0.361325 0.932440i \(-0.382325\pi\)
0.361325 + 0.932440i \(0.382325\pi\)
\(14\) 2.30278 0.615443
\(15\) 0 0
\(16\) 0.302776 0.0756939
\(17\) −4.60555 −1.11701 −0.558505 0.829501i \(-0.688625\pi\)
−0.558505 + 0.829501i \(0.688625\pi\)
\(18\) −2.30278 −0.542769
\(19\) 6.60555 1.51542 0.757709 0.652593i \(-0.226319\pi\)
0.757709 + 0.652593i \(0.226319\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 6.90833 1.47286
\(23\) 6.21110 1.29510 0.647552 0.762021i \(-0.275793\pi\)
0.647552 + 0.762021i \(0.275793\pi\)
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) −6.00000 −1.17670
\(27\) −1.00000 −0.192450
\(28\) −3.30278 −0.624166
\(29\) −7.60555 −1.41232 −0.706158 0.708055i \(-0.749573\pi\)
−0.706158 + 0.708055i \(0.749573\pi\)
\(30\) 0 0
\(31\) −7.21110 −1.29515 −0.647576 0.762001i \(-0.724217\pi\)
−0.647576 + 0.762001i \(0.724217\pi\)
\(32\) 5.30278 0.937407
\(33\) 3.00000 0.522233
\(34\) 10.6056 1.81884
\(35\) 0 0
\(36\) 3.30278 0.550463
\(37\) 4.21110 0.692301 0.346150 0.938179i \(-0.387489\pi\)
0.346150 + 0.938179i \(0.387489\pi\)
\(38\) −15.2111 −2.46757
\(39\) −2.60555 −0.417222
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −2.30278 −0.355326
\(43\) −9.60555 −1.46483 −0.732416 0.680857i \(-0.761608\pi\)
−0.732416 + 0.680857i \(0.761608\pi\)
\(44\) −9.90833 −1.49374
\(45\) 0 0
\(46\) −14.3028 −2.10883
\(47\) −10.6056 −1.54698 −0.773489 0.633809i \(-0.781490\pi\)
−0.773489 + 0.633809i \(0.781490\pi\)
\(48\) −0.302776 −0.0437019
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.60555 0.644906
\(52\) 8.60555 1.19338
\(53\) −3.21110 −0.441079 −0.220539 0.975378i \(-0.570782\pi\)
−0.220539 + 0.975378i \(0.570782\pi\)
\(54\) 2.30278 0.313368
\(55\) 0 0
\(56\) 3.00000 0.400892
\(57\) −6.60555 −0.874927
\(58\) 17.5139 2.29968
\(59\) −10.6056 −1.38073 −0.690363 0.723464i \(-0.742549\pi\)
−0.690363 + 0.723464i \(0.742549\pi\)
\(60\) 0 0
\(61\) −1.21110 −0.155066 −0.0775329 0.996990i \(-0.524704\pi\)
−0.0775329 + 0.996990i \(0.524704\pi\)
\(62\) 16.6056 2.10891
\(63\) −1.00000 −0.125988
\(64\) −12.8167 −1.60208
\(65\) 0 0
\(66\) −6.90833 −0.850356
\(67\) −15.6056 −1.90652 −0.953261 0.302149i \(-0.902296\pi\)
−0.953261 + 0.302149i \(0.902296\pi\)
\(68\) −15.2111 −1.84462
\(69\) −6.21110 −0.747729
\(70\) 0 0
\(71\) −3.00000 −0.356034 −0.178017 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) −3.00000 −0.353553
\(73\) −0.605551 −0.0708744 −0.0354372 0.999372i \(-0.511282\pi\)
−0.0354372 + 0.999372i \(0.511282\pi\)
\(74\) −9.69722 −1.12728
\(75\) 0 0
\(76\) 21.8167 2.50254
\(77\) 3.00000 0.341882
\(78\) 6.00000 0.679366
\(79\) −14.8167 −1.66700 −0.833502 0.552517i \(-0.813668\pi\)
−0.833502 + 0.552517i \(0.813668\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.21110 0.352464 0.176232 0.984349i \(-0.443609\pi\)
0.176232 + 0.984349i \(0.443609\pi\)
\(84\) 3.30278 0.360362
\(85\) 0 0
\(86\) 22.1194 2.38520
\(87\) 7.60555 0.815401
\(88\) 9.00000 0.959403
\(89\) 7.81665 0.828564 0.414282 0.910149i \(-0.364033\pi\)
0.414282 + 0.910149i \(0.364033\pi\)
\(90\) 0 0
\(91\) −2.60555 −0.273136
\(92\) 20.5139 2.13872
\(93\) 7.21110 0.747757
\(94\) 24.4222 2.51896
\(95\) 0 0
\(96\) −5.30278 −0.541212
\(97\) 0.788897 0.0801004 0.0400502 0.999198i \(-0.487248\pi\)
0.0400502 + 0.999198i \(0.487248\pi\)
\(98\) −2.30278 −0.232615
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) 16.6056 1.65231 0.826157 0.563440i \(-0.190522\pi\)
0.826157 + 0.563440i \(0.190522\pi\)
\(102\) −10.6056 −1.05011
\(103\) −3.81665 −0.376066 −0.188033 0.982163i \(-0.560211\pi\)
−0.188033 + 0.982163i \(0.560211\pi\)
\(104\) −7.81665 −0.766486
\(105\) 0 0
\(106\) 7.39445 0.718212
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −3.30278 −0.317810
\(109\) 2.21110 0.211785 0.105893 0.994378i \(-0.466230\pi\)
0.105893 + 0.994378i \(0.466230\pi\)
\(110\) 0 0
\(111\) −4.21110 −0.399700
\(112\) −0.302776 −0.0286096
\(113\) 10.8167 1.01755 0.508773 0.860901i \(-0.330099\pi\)
0.508773 + 0.860901i \(0.330099\pi\)
\(114\) 15.2111 1.42465
\(115\) 0 0
\(116\) −25.1194 −2.33228
\(117\) 2.60555 0.240883
\(118\) 24.4222 2.24825
\(119\) 4.60555 0.422190
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 2.78890 0.252495
\(123\) 0 0
\(124\) −23.8167 −2.13880
\(125\) 0 0
\(126\) 2.30278 0.205148
\(127\) −12.8167 −1.13729 −0.568647 0.822582i \(-0.692533\pi\)
−0.568647 + 0.822582i \(0.692533\pi\)
\(128\) 18.9083 1.67128
\(129\) 9.60555 0.845722
\(130\) 0 0
\(131\) −7.39445 −0.646056 −0.323028 0.946389i \(-0.604701\pi\)
−0.323028 + 0.946389i \(0.604701\pi\)
\(132\) 9.90833 0.862409
\(133\) −6.60555 −0.572774
\(134\) 35.9361 3.10440
\(135\) 0 0
\(136\) 13.8167 1.18477
\(137\) 3.21110 0.274343 0.137172 0.990547i \(-0.456199\pi\)
0.137172 + 0.990547i \(0.456199\pi\)
\(138\) 14.3028 1.21753
\(139\) 19.0278 1.61391 0.806957 0.590611i \(-0.201113\pi\)
0.806957 + 0.590611i \(0.201113\pi\)
\(140\) 0 0
\(141\) 10.6056 0.893149
\(142\) 6.90833 0.579734
\(143\) −7.81665 −0.653661
\(144\) 0.302776 0.0252313
\(145\) 0 0
\(146\) 1.39445 0.115405
\(147\) −1.00000 −0.0824786
\(148\) 13.9083 1.14326
\(149\) −16.3944 −1.34309 −0.671543 0.740966i \(-0.734368\pi\)
−0.671543 + 0.740966i \(0.734368\pi\)
\(150\) 0 0
\(151\) 6.81665 0.554731 0.277366 0.960764i \(-0.410539\pi\)
0.277366 + 0.960764i \(0.410539\pi\)
\(152\) −19.8167 −1.60734
\(153\) −4.60555 −0.372337
\(154\) −6.90833 −0.556689
\(155\) 0 0
\(156\) −8.60555 −0.688996
\(157\) −14.4222 −1.15102 −0.575509 0.817796i \(-0.695196\pi\)
−0.575509 + 0.817796i \(0.695196\pi\)
\(158\) 34.1194 2.71440
\(159\) 3.21110 0.254657
\(160\) 0 0
\(161\) −6.21110 −0.489503
\(162\) −2.30278 −0.180923
\(163\) −17.2111 −1.34808 −0.674039 0.738696i \(-0.735442\pi\)
−0.674039 + 0.738696i \(0.735442\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −7.39445 −0.573921
\(167\) 17.0278 1.31765 0.658824 0.752297i \(-0.271054\pi\)
0.658824 + 0.752297i \(0.271054\pi\)
\(168\) −3.00000 −0.231455
\(169\) −6.21110 −0.477777
\(170\) 0 0
\(171\) 6.60555 0.505139
\(172\) −31.7250 −2.41901
\(173\) 7.81665 0.594289 0.297145 0.954832i \(-0.403966\pi\)
0.297145 + 0.954832i \(0.403966\pi\)
\(174\) −17.5139 −1.32772
\(175\) 0 0
\(176\) −0.908327 −0.0684677
\(177\) 10.6056 0.797162
\(178\) −18.0000 −1.34916
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 12.6056 0.936963 0.468482 0.883473i \(-0.344801\pi\)
0.468482 + 0.883473i \(0.344801\pi\)
\(182\) 6.00000 0.444750
\(183\) 1.21110 0.0895273
\(184\) −18.6333 −1.37367
\(185\) 0 0
\(186\) −16.6056 −1.21758
\(187\) 13.8167 1.01037
\(188\) −35.0278 −2.55466
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −2.78890 −0.201798 −0.100899 0.994897i \(-0.532172\pi\)
−0.100899 + 0.994897i \(0.532172\pi\)
\(192\) 12.8167 0.924962
\(193\) −8.21110 −0.591048 −0.295524 0.955335i \(-0.595494\pi\)
−0.295524 + 0.955335i \(0.595494\pi\)
\(194\) −1.81665 −0.130428
\(195\) 0 0
\(196\) 3.30278 0.235913
\(197\) −16.8167 −1.19814 −0.599068 0.800698i \(-0.704462\pi\)
−0.599068 + 0.800698i \(0.704462\pi\)
\(198\) 6.90833 0.490953
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) 15.6056 1.10073
\(202\) −38.2389 −2.69048
\(203\) 7.60555 0.533805
\(204\) 15.2111 1.06499
\(205\) 0 0
\(206\) 8.78890 0.612352
\(207\) 6.21110 0.431701
\(208\) 0.788897 0.0547002
\(209\) −19.8167 −1.37075
\(210\) 0 0
\(211\) 26.4222 1.81898 0.909490 0.415726i \(-0.136473\pi\)
0.909490 + 0.415726i \(0.136473\pi\)
\(212\) −10.6056 −0.728392
\(213\) 3.00000 0.205557
\(214\) 0 0
\(215\) 0 0
\(216\) 3.00000 0.204124
\(217\) 7.21110 0.489522
\(218\) −5.09167 −0.344852
\(219\) 0.605551 0.0409194
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 9.69722 0.650835
\(223\) −15.3944 −1.03089 −0.515444 0.856923i \(-0.672373\pi\)
−0.515444 + 0.856923i \(0.672373\pi\)
\(224\) −5.30278 −0.354307
\(225\) 0 0
\(226\) −24.9083 −1.65688
\(227\) −22.6056 −1.50038 −0.750192 0.661221i \(-0.770039\pi\)
−0.750192 + 0.661221i \(0.770039\pi\)
\(228\) −21.8167 −1.44484
\(229\) −7.21110 −0.476523 −0.238262 0.971201i \(-0.576578\pi\)
−0.238262 + 0.971201i \(0.576578\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 22.8167 1.49799
\(233\) 1.18335 0.0775236 0.0387618 0.999248i \(-0.487659\pi\)
0.0387618 + 0.999248i \(0.487659\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) −35.0278 −2.28011
\(237\) 14.8167 0.962445
\(238\) −10.6056 −0.687456
\(239\) 14.7889 0.956614 0.478307 0.878193i \(-0.341251\pi\)
0.478307 + 0.878193i \(0.341251\pi\)
\(240\) 0 0
\(241\) 9.39445 0.605150 0.302575 0.953126i \(-0.402154\pi\)
0.302575 + 0.953126i \(0.402154\pi\)
\(242\) 4.60555 0.296056
\(243\) −1.00000 −0.0641500
\(244\) −4.00000 −0.256074
\(245\) 0 0
\(246\) 0 0
\(247\) 17.2111 1.09512
\(248\) 21.6333 1.37372
\(249\) −3.21110 −0.203495
\(250\) 0 0
\(251\) −13.8167 −0.872099 −0.436050 0.899923i \(-0.643623\pi\)
−0.436050 + 0.899923i \(0.643623\pi\)
\(252\) −3.30278 −0.208055
\(253\) −18.6333 −1.17147
\(254\) 29.5139 1.85187
\(255\) 0 0
\(256\) −17.9083 −1.11927
\(257\) 21.6333 1.34945 0.674724 0.738070i \(-0.264263\pi\)
0.674724 + 0.738070i \(0.264263\pi\)
\(258\) −22.1194 −1.37710
\(259\) −4.21110 −0.261665
\(260\) 0 0
\(261\) −7.60555 −0.470772
\(262\) 17.0278 1.05198
\(263\) 5.78890 0.356959 0.178479 0.983944i \(-0.442882\pi\)
0.178479 + 0.983944i \(0.442882\pi\)
\(264\) −9.00000 −0.553912
\(265\) 0 0
\(266\) 15.2111 0.932653
\(267\) −7.81665 −0.478371
\(268\) −51.5416 −3.14841
\(269\) −3.21110 −0.195784 −0.0978922 0.995197i \(-0.531210\pi\)
−0.0978922 + 0.995197i \(0.531210\pi\)
\(270\) 0 0
\(271\) −26.6056 −1.61617 −0.808086 0.589064i \(-0.799496\pi\)
−0.808086 + 0.589064i \(0.799496\pi\)
\(272\) −1.39445 −0.0845509
\(273\) 2.60555 0.157695
\(274\) −7.39445 −0.446715
\(275\) 0 0
\(276\) −20.5139 −1.23479
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) −43.8167 −2.62795
\(279\) −7.21110 −0.431717
\(280\) 0 0
\(281\) −1.18335 −0.0705925 −0.0352963 0.999377i \(-0.511237\pi\)
−0.0352963 + 0.999377i \(0.511237\pi\)
\(282\) −24.4222 −1.45432
\(283\) 13.6333 0.810416 0.405208 0.914225i \(-0.367199\pi\)
0.405208 + 0.914225i \(0.367199\pi\)
\(284\) −9.90833 −0.587951
\(285\) 0 0
\(286\) 18.0000 1.06436
\(287\) 0 0
\(288\) 5.30278 0.312469
\(289\) 4.21110 0.247712
\(290\) 0 0
\(291\) −0.788897 −0.0462460
\(292\) −2.00000 −0.117041
\(293\) −10.6056 −0.619583 −0.309791 0.950805i \(-0.600259\pi\)
−0.309791 + 0.950805i \(0.600259\pi\)
\(294\) 2.30278 0.134301
\(295\) 0 0
\(296\) −12.6333 −0.734296
\(297\) 3.00000 0.174078
\(298\) 37.7527 2.18696
\(299\) 16.1833 0.935907
\(300\) 0 0
\(301\) 9.60555 0.553655
\(302\) −15.6972 −0.903274
\(303\) −16.6056 −0.953964
\(304\) 2.00000 0.114708
\(305\) 0 0
\(306\) 10.6056 0.606279
\(307\) 8.60555 0.491145 0.245572 0.969378i \(-0.421024\pi\)
0.245572 + 0.969378i \(0.421024\pi\)
\(308\) 9.90833 0.564579
\(309\) 3.81665 0.217122
\(310\) 0 0
\(311\) 13.8167 0.783471 0.391735 0.920078i \(-0.371875\pi\)
0.391735 + 0.920078i \(0.371875\pi\)
\(312\) 7.81665 0.442531
\(313\) 23.8167 1.34620 0.673098 0.739553i \(-0.264963\pi\)
0.673098 + 0.739553i \(0.264963\pi\)
\(314\) 33.2111 1.87421
\(315\) 0 0
\(316\) −48.9361 −2.75287
\(317\) 16.3944 0.920804 0.460402 0.887711i \(-0.347705\pi\)
0.460402 + 0.887711i \(0.347705\pi\)
\(318\) −7.39445 −0.414660
\(319\) 22.8167 1.27749
\(320\) 0 0
\(321\) 0 0
\(322\) 14.3028 0.797063
\(323\) −30.4222 −1.69274
\(324\) 3.30278 0.183488
\(325\) 0 0
\(326\) 39.6333 2.19509
\(327\) −2.21110 −0.122274
\(328\) 0 0
\(329\) 10.6056 0.584703
\(330\) 0 0
\(331\) −21.2389 −1.16739 −0.583697 0.811972i \(-0.698394\pi\)
−0.583697 + 0.811972i \(0.698394\pi\)
\(332\) 10.6056 0.582055
\(333\) 4.21110 0.230767
\(334\) −39.2111 −2.14554
\(335\) 0 0
\(336\) 0.302776 0.0165178
\(337\) 7.21110 0.392814 0.196407 0.980522i \(-0.437073\pi\)
0.196407 + 0.980522i \(0.437073\pi\)
\(338\) 14.3028 0.777968
\(339\) −10.8167 −0.587480
\(340\) 0 0
\(341\) 21.6333 1.17151
\(342\) −15.2111 −0.822522
\(343\) −1.00000 −0.0539949
\(344\) 28.8167 1.55369
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) −30.2111 −1.62182 −0.810908 0.585173i \(-0.801027\pi\)
−0.810908 + 0.585173i \(0.801027\pi\)
\(348\) 25.1194 1.34654
\(349\) 31.4500 1.68348 0.841739 0.539885i \(-0.181532\pi\)
0.841739 + 0.539885i \(0.181532\pi\)
\(350\) 0 0
\(351\) −2.60555 −0.139074
\(352\) −15.9083 −0.847917
\(353\) 30.0000 1.59674 0.798369 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) −24.4222 −1.29803
\(355\) 0 0
\(356\) 25.8167 1.36828
\(357\) −4.60555 −0.243752
\(358\) 0 0
\(359\) 24.6333 1.30010 0.650048 0.759893i \(-0.274749\pi\)
0.650048 + 0.759893i \(0.274749\pi\)
\(360\) 0 0
\(361\) 24.6333 1.29649
\(362\) −29.0278 −1.52567
\(363\) 2.00000 0.104973
\(364\) −8.60555 −0.451053
\(365\) 0 0
\(366\) −2.78890 −0.145778
\(367\) −14.4222 −0.752833 −0.376416 0.926451i \(-0.622844\pi\)
−0.376416 + 0.926451i \(0.622844\pi\)
\(368\) 1.88057 0.0980315
\(369\) 0 0
\(370\) 0 0
\(371\) 3.21110 0.166712
\(372\) 23.8167 1.23484
\(373\) 1.00000 0.0517780 0.0258890 0.999665i \(-0.491758\pi\)
0.0258890 + 0.999665i \(0.491758\pi\)
\(374\) −31.8167 −1.64520
\(375\) 0 0
\(376\) 31.8167 1.64082
\(377\) −19.8167 −1.02061
\(378\) −2.30278 −0.118442
\(379\) −14.8167 −0.761080 −0.380540 0.924764i \(-0.624262\pi\)
−0.380540 + 0.924764i \(0.624262\pi\)
\(380\) 0 0
\(381\) 12.8167 0.656617
\(382\) 6.42221 0.328589
\(383\) 26.2389 1.34074 0.670372 0.742026i \(-0.266135\pi\)
0.670372 + 0.742026i \(0.266135\pi\)
\(384\) −18.9083 −0.964912
\(385\) 0 0
\(386\) 18.9083 0.962408
\(387\) −9.60555 −0.488278
\(388\) 2.60555 0.132277
\(389\) −28.8167 −1.46106 −0.730531 0.682879i \(-0.760727\pi\)
−0.730531 + 0.682879i \(0.760727\pi\)
\(390\) 0 0
\(391\) −28.6056 −1.44664
\(392\) −3.00000 −0.151523
\(393\) 7.39445 0.373001
\(394\) 38.7250 1.95094
\(395\) 0 0
\(396\) −9.90833 −0.497912
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) −18.4222 −0.923422
\(399\) 6.60555 0.330691
\(400\) 0 0
\(401\) −16.8167 −0.839784 −0.419892 0.907574i \(-0.637932\pi\)
−0.419892 + 0.907574i \(0.637932\pi\)
\(402\) −35.9361 −1.79233
\(403\) −18.7889 −0.935942
\(404\) 54.8444 2.72861
\(405\) 0 0
\(406\) −17.5139 −0.869199
\(407\) −12.6333 −0.626210
\(408\) −13.8167 −0.684026
\(409\) −11.8167 −0.584296 −0.292148 0.956373i \(-0.594370\pi\)
−0.292148 + 0.956373i \(0.594370\pi\)
\(410\) 0 0
\(411\) −3.21110 −0.158392
\(412\) −12.6056 −0.621031
\(413\) 10.6056 0.521865
\(414\) −14.3028 −0.702943
\(415\) 0 0
\(416\) 13.8167 0.677417
\(417\) −19.0278 −0.931793
\(418\) 45.6333 2.23200
\(419\) 6.00000 0.293119 0.146560 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 0 0
\(421\) −10.2111 −0.497659 −0.248829 0.968547i \(-0.580046\pi\)
−0.248829 + 0.968547i \(0.580046\pi\)
\(422\) −60.8444 −2.96186
\(423\) −10.6056 −0.515660
\(424\) 9.63331 0.467835
\(425\) 0 0
\(426\) −6.90833 −0.334710
\(427\) 1.21110 0.0586094
\(428\) 0 0
\(429\) 7.81665 0.377392
\(430\) 0 0
\(431\) 5.57779 0.268673 0.134336 0.990936i \(-0.457110\pi\)
0.134336 + 0.990936i \(0.457110\pi\)
\(432\) −0.302776 −0.0145673
\(433\) 37.2111 1.78825 0.894126 0.447816i \(-0.147798\pi\)
0.894126 + 0.447816i \(0.147798\pi\)
\(434\) −16.6056 −0.797092
\(435\) 0 0
\(436\) 7.30278 0.349740
\(437\) 41.0278 1.96262
\(438\) −1.39445 −0.0666293
\(439\) 6.60555 0.315266 0.157633 0.987498i \(-0.449614\pi\)
0.157633 + 0.987498i \(0.449614\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 27.6333 1.31438
\(443\) −15.6333 −0.742761 −0.371380 0.928481i \(-0.621115\pi\)
−0.371380 + 0.928481i \(0.621115\pi\)
\(444\) −13.9083 −0.660060
\(445\) 0 0
\(446\) 35.4500 1.67860
\(447\) 16.3944 0.775431
\(448\) 12.8167 0.605530
\(449\) −34.8167 −1.64310 −0.821550 0.570137i \(-0.806890\pi\)
−0.821550 + 0.570137i \(0.806890\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 35.7250 1.68036
\(453\) −6.81665 −0.320274
\(454\) 52.0555 2.44309
\(455\) 0 0
\(456\) 19.8167 0.928000
\(457\) 3.78890 0.177237 0.0886186 0.996066i \(-0.471755\pi\)
0.0886186 + 0.996066i \(0.471755\pi\)
\(458\) 16.6056 0.775926
\(459\) 4.60555 0.214969
\(460\) 0 0
\(461\) −26.2389 −1.22207 −0.611033 0.791605i \(-0.709246\pi\)
−0.611033 + 0.791605i \(0.709246\pi\)
\(462\) 6.90833 0.321404
\(463\) −17.2111 −0.799868 −0.399934 0.916544i \(-0.630967\pi\)
−0.399934 + 0.916544i \(0.630967\pi\)
\(464\) −2.30278 −0.106904
\(465\) 0 0
\(466\) −2.72498 −0.126232
\(467\) 0.422205 0.0195373 0.00976866 0.999952i \(-0.496890\pi\)
0.00976866 + 0.999952i \(0.496890\pi\)
\(468\) 8.60555 0.397792
\(469\) 15.6056 0.720597
\(470\) 0 0
\(471\) 14.4222 0.664540
\(472\) 31.8167 1.46448
\(473\) 28.8167 1.32499
\(474\) −34.1194 −1.56716
\(475\) 0 0
\(476\) 15.2111 0.697200
\(477\) −3.21110 −0.147026
\(478\) −34.0555 −1.55766
\(479\) 34.6056 1.58117 0.790584 0.612354i \(-0.209777\pi\)
0.790584 + 0.612354i \(0.209777\pi\)
\(480\) 0 0
\(481\) 10.9722 0.500291
\(482\) −21.6333 −0.985370
\(483\) 6.21110 0.282615
\(484\) −6.60555 −0.300252
\(485\) 0 0
\(486\) 2.30278 0.104456
\(487\) 20.8167 0.943293 0.471646 0.881788i \(-0.343660\pi\)
0.471646 + 0.881788i \(0.343660\pi\)
\(488\) 3.63331 0.164472
\(489\) 17.2111 0.778313
\(490\) 0 0
\(491\) −3.00000 −0.135388 −0.0676941 0.997706i \(-0.521564\pi\)
−0.0676941 + 0.997706i \(0.521564\pi\)
\(492\) 0 0
\(493\) 35.0278 1.57757
\(494\) −39.6333 −1.78319
\(495\) 0 0
\(496\) −2.18335 −0.0980351
\(497\) 3.00000 0.134568
\(498\) 7.39445 0.331353
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 0 0
\(501\) −17.0278 −0.760744
\(502\) 31.8167 1.42005
\(503\) 8.78890 0.391878 0.195939 0.980616i \(-0.437225\pi\)
0.195939 + 0.980616i \(0.437225\pi\)
\(504\) 3.00000 0.133631
\(505\) 0 0
\(506\) 42.9083 1.90751
\(507\) 6.21110 0.275845
\(508\) −42.3305 −1.87811
\(509\) 1.39445 0.0618079 0.0309039 0.999522i \(-0.490161\pi\)
0.0309039 + 0.999522i \(0.490161\pi\)
\(510\) 0 0
\(511\) 0.605551 0.0267880
\(512\) 3.42221 0.151242
\(513\) −6.60555 −0.291642
\(514\) −49.8167 −2.19732
\(515\) 0 0
\(516\) 31.7250 1.39661
\(517\) 31.8167 1.39929
\(518\) 9.69722 0.426072
\(519\) −7.81665 −0.343113
\(520\) 0 0
\(521\) −36.4222 −1.59569 −0.797843 0.602865i \(-0.794026\pi\)
−0.797843 + 0.602865i \(0.794026\pi\)
\(522\) 17.5139 0.766562
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) −24.4222 −1.06689
\(525\) 0 0
\(526\) −13.3305 −0.581239
\(527\) 33.2111 1.44670
\(528\) 0.908327 0.0395299
\(529\) 15.5778 0.677295
\(530\) 0 0
\(531\) −10.6056 −0.460242
\(532\) −21.8167 −0.945872
\(533\) 0 0
\(534\) 18.0000 0.778936
\(535\) 0 0
\(536\) 46.8167 2.02217
\(537\) 0 0
\(538\) 7.39445 0.318797
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) −31.4222 −1.35095 −0.675473 0.737385i \(-0.736061\pi\)
−0.675473 + 0.737385i \(0.736061\pi\)
\(542\) 61.2666 2.63163
\(543\) −12.6056 −0.540956
\(544\) −24.4222 −1.04709
\(545\) 0 0
\(546\) −6.00000 −0.256776
\(547\) 29.6056 1.26584 0.632921 0.774216i \(-0.281856\pi\)
0.632921 + 0.774216i \(0.281856\pi\)
\(548\) 10.6056 0.453047
\(549\) −1.21110 −0.0516886
\(550\) 0 0
\(551\) −50.2389 −2.14025
\(552\) 18.6333 0.793086
\(553\) 14.8167 0.630068
\(554\) −23.0278 −0.978356
\(555\) 0 0
\(556\) 62.8444 2.66520
\(557\) −11.2389 −0.476206 −0.238103 0.971240i \(-0.576526\pi\)
−0.238103 + 0.971240i \(0.576526\pi\)
\(558\) 16.6056 0.702969
\(559\) −25.0278 −1.05856
\(560\) 0 0
\(561\) −13.8167 −0.583340
\(562\) 2.72498 0.114946
\(563\) −15.2111 −0.641072 −0.320536 0.947236i \(-0.603863\pi\)
−0.320536 + 0.947236i \(0.603863\pi\)
\(564\) 35.0278 1.47493
\(565\) 0 0
\(566\) −31.3944 −1.31961
\(567\) −1.00000 −0.0419961
\(568\) 9.00000 0.377632
\(569\) −1.18335 −0.0496085 −0.0248042 0.999692i \(-0.507896\pi\)
−0.0248042 + 0.999692i \(0.507896\pi\)
\(570\) 0 0
\(571\) −5.60555 −0.234585 −0.117293 0.993097i \(-0.537422\pi\)
−0.117293 + 0.993097i \(0.537422\pi\)
\(572\) −25.8167 −1.07945
\(573\) 2.78890 0.116508
\(574\) 0 0
\(575\) 0 0
\(576\) −12.8167 −0.534027
\(577\) −30.6056 −1.27413 −0.637063 0.770812i \(-0.719851\pi\)
−0.637063 + 0.770812i \(0.719851\pi\)
\(578\) −9.69722 −0.403351
\(579\) 8.21110 0.341242
\(580\) 0 0
\(581\) −3.21110 −0.133219
\(582\) 1.81665 0.0753027
\(583\) 9.63331 0.398971
\(584\) 1.81665 0.0751737
\(585\) 0 0
\(586\) 24.4222 1.00887
\(587\) −37.8167 −1.56086 −0.780430 0.625243i \(-0.785000\pi\)
−0.780430 + 0.625243i \(0.785000\pi\)
\(588\) −3.30278 −0.136204
\(589\) −47.6333 −1.96270
\(590\) 0 0
\(591\) 16.8167 0.691745
\(592\) 1.27502 0.0524030
\(593\) −27.6333 −1.13476 −0.567382 0.823455i \(-0.692044\pi\)
−0.567382 + 0.823455i \(0.692044\pi\)
\(594\) −6.90833 −0.283452
\(595\) 0 0
\(596\) −54.1472 −2.21796
\(597\) −8.00000 −0.327418
\(598\) −37.2666 −1.52395
\(599\) 5.78890 0.236528 0.118264 0.992982i \(-0.462267\pi\)
0.118264 + 0.992982i \(0.462267\pi\)
\(600\) 0 0
\(601\) 17.2111 0.702056 0.351028 0.936365i \(-0.385832\pi\)
0.351028 + 0.936365i \(0.385832\pi\)
\(602\) −22.1194 −0.901521
\(603\) −15.6056 −0.635507
\(604\) 22.5139 0.916077
\(605\) 0 0
\(606\) 38.2389 1.55335
\(607\) 23.8167 0.966688 0.483344 0.875430i \(-0.339422\pi\)
0.483344 + 0.875430i \(0.339422\pi\)
\(608\) 35.0278 1.42056
\(609\) −7.60555 −0.308192
\(610\) 0 0
\(611\) −27.6333 −1.11792
\(612\) −15.2111 −0.614872
\(613\) 49.4222 1.99614 0.998072 0.0620663i \(-0.0197690\pi\)
0.998072 + 0.0620663i \(0.0197690\pi\)
\(614\) −19.8167 −0.799735
\(615\) 0 0
\(616\) −9.00000 −0.362620
\(617\) −22.8167 −0.918564 −0.459282 0.888291i \(-0.651893\pi\)
−0.459282 + 0.888291i \(0.651893\pi\)
\(618\) −8.78890 −0.353541
\(619\) −2.60555 −0.104726 −0.0523630 0.998628i \(-0.516675\pi\)
−0.0523630 + 0.998628i \(0.516675\pi\)
\(620\) 0 0
\(621\) −6.21110 −0.249243
\(622\) −31.8167 −1.27573
\(623\) −7.81665 −0.313168
\(624\) −0.788897 −0.0315812
\(625\) 0 0
\(626\) −54.8444 −2.19202
\(627\) 19.8167 0.791401
\(628\) −47.6333 −1.90078
\(629\) −19.3944 −0.773307
\(630\) 0 0
\(631\) 22.0278 0.876911 0.438456 0.898753i \(-0.355526\pi\)
0.438456 + 0.898753i \(0.355526\pi\)
\(632\) 44.4500 1.76812
\(633\) −26.4222 −1.05019
\(634\) −37.7527 −1.49935
\(635\) 0 0
\(636\) 10.6056 0.420537
\(637\) 2.60555 0.103236
\(638\) −52.5416 −2.08014
\(639\) −3.00000 −0.118678
\(640\) 0 0
\(641\) 22.8167 0.901204 0.450602 0.892725i \(-0.351209\pi\)
0.450602 + 0.892725i \(0.351209\pi\)
\(642\) 0 0
\(643\) −38.4222 −1.51522 −0.757612 0.652705i \(-0.773634\pi\)
−0.757612 + 0.652705i \(0.773634\pi\)
\(644\) −20.5139 −0.808360
\(645\) 0 0
\(646\) 70.0555 2.75630
\(647\) −6.00000 −0.235884 −0.117942 0.993020i \(-0.537630\pi\)
−0.117942 + 0.993020i \(0.537630\pi\)
\(648\) −3.00000 −0.117851
\(649\) 31.8167 1.24891
\(650\) 0 0
\(651\) −7.21110 −0.282625
\(652\) −56.8444 −2.22620
\(653\) 3.21110 0.125660 0.0628301 0.998024i \(-0.479987\pi\)
0.0628301 + 0.998024i \(0.479987\pi\)
\(654\) 5.09167 0.199100
\(655\) 0 0
\(656\) 0 0
\(657\) −0.605551 −0.0236248
\(658\) −24.4222 −0.952077
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) −20.1833 −0.785041 −0.392521 0.919743i \(-0.628397\pi\)
−0.392521 + 0.919743i \(0.628397\pi\)
\(662\) 48.9083 1.90088
\(663\) 12.0000 0.466041
\(664\) −9.63331 −0.373845
\(665\) 0 0
\(666\) −9.69722 −0.375760
\(667\) −47.2389 −1.82910
\(668\) 56.2389 2.17595
\(669\) 15.3944 0.595184
\(670\) 0 0
\(671\) 3.63331 0.140262
\(672\) 5.30278 0.204559
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) −16.6056 −0.639622
\(675\) 0 0
\(676\) −20.5139 −0.788995
\(677\) 8.78890 0.337785 0.168892 0.985634i \(-0.445981\pi\)
0.168892 + 0.985634i \(0.445981\pi\)
\(678\) 24.9083 0.956599
\(679\) −0.788897 −0.0302751
\(680\) 0 0
\(681\) 22.6056 0.866247
\(682\) −49.8167 −1.90758
\(683\) 18.6333 0.712984 0.356492 0.934298i \(-0.383973\pi\)
0.356492 + 0.934298i \(0.383973\pi\)
\(684\) 21.8167 0.834181
\(685\) 0 0
\(686\) 2.30278 0.0879204
\(687\) 7.21110 0.275121
\(688\) −2.90833 −0.110879
\(689\) −8.36669 −0.318746
\(690\) 0 0
\(691\) 41.2111 1.56774 0.783872 0.620922i \(-0.213242\pi\)
0.783872 + 0.620922i \(0.213242\pi\)
\(692\) 25.8167 0.981402
\(693\) 3.00000 0.113961
\(694\) 69.5694 2.64082
\(695\) 0 0
\(696\) −22.8167 −0.864863
\(697\) 0 0
\(698\) −72.4222 −2.74122
\(699\) −1.18335 −0.0447583
\(700\) 0 0
\(701\) −20.7889 −0.785186 −0.392593 0.919712i \(-0.628422\pi\)
−0.392593 + 0.919712i \(0.628422\pi\)
\(702\) 6.00000 0.226455
\(703\) 27.8167 1.04912
\(704\) 38.4500 1.44914
\(705\) 0 0
\(706\) −69.0833 −2.59998
\(707\) −16.6056 −0.624516
\(708\) 35.0278 1.31642
\(709\) −34.8444 −1.30861 −0.654305 0.756231i \(-0.727039\pi\)
−0.654305 + 0.756231i \(0.727039\pi\)
\(710\) 0 0
\(711\) −14.8167 −0.555668
\(712\) −23.4500 −0.878824
\(713\) −44.7889 −1.67736
\(714\) 10.6056 0.396903
\(715\) 0 0
\(716\) 0 0
\(717\) −14.7889 −0.552301
\(718\) −56.7250 −2.11696
\(719\) 26.7889 0.999057 0.499529 0.866297i \(-0.333507\pi\)
0.499529 + 0.866297i \(0.333507\pi\)
\(720\) 0 0
\(721\) 3.81665 0.142140
\(722\) −56.7250 −2.11109
\(723\) −9.39445 −0.349383
\(724\) 41.6333 1.54729
\(725\) 0 0
\(726\) −4.60555 −0.170928
\(727\) 23.3944 0.867652 0.433826 0.900997i \(-0.357163\pi\)
0.433826 + 0.900997i \(0.357163\pi\)
\(728\) 7.81665 0.289704
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 44.2389 1.63623
\(732\) 4.00000 0.147844
\(733\) −20.0000 −0.738717 −0.369358 0.929287i \(-0.620423\pi\)
−0.369358 + 0.929287i \(0.620423\pi\)
\(734\) 33.2111 1.22584
\(735\) 0 0
\(736\) 32.9361 1.21404
\(737\) 46.8167 1.72451
\(738\) 0 0
\(739\) 3.18335 0.117101 0.0585506 0.998284i \(-0.481352\pi\)
0.0585506 + 0.998284i \(0.481352\pi\)
\(740\) 0 0
\(741\) −17.2111 −0.632266
\(742\) −7.39445 −0.271459
\(743\) 27.6333 1.01377 0.506884 0.862014i \(-0.330797\pi\)
0.506884 + 0.862014i \(0.330797\pi\)
\(744\) −21.6333 −0.793116
\(745\) 0 0
\(746\) −2.30278 −0.0843106
\(747\) 3.21110 0.117488
\(748\) 45.6333 1.66852
\(749\) 0 0
\(750\) 0 0
\(751\) −0.366692 −0.0133808 −0.00669040 0.999978i \(-0.502130\pi\)
−0.00669040 + 0.999978i \(0.502130\pi\)
\(752\) −3.21110 −0.117097
\(753\) 13.8167 0.503507
\(754\) 45.6333 1.66187
\(755\) 0 0
\(756\) 3.30278 0.120121
\(757\) −41.0000 −1.49017 −0.745085 0.666969i \(-0.767591\pi\)
−0.745085 + 0.666969i \(0.767591\pi\)
\(758\) 34.1194 1.23927
\(759\) 18.6333 0.676346
\(760\) 0 0
\(761\) −27.6333 −1.00171 −0.500853 0.865532i \(-0.666980\pi\)
−0.500853 + 0.865532i \(0.666980\pi\)
\(762\) −29.5139 −1.06917
\(763\) −2.21110 −0.0800473
\(764\) −9.21110 −0.333246
\(765\) 0 0
\(766\) −60.4222 −2.18314
\(767\) −27.6333 −0.997781
\(768\) 17.9083 0.646211
\(769\) −31.6333 −1.14073 −0.570363 0.821393i \(-0.693198\pi\)
−0.570363 + 0.821393i \(0.693198\pi\)
\(770\) 0 0
\(771\) −21.6333 −0.779105
\(772\) −27.1194 −0.976050
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 22.1194 0.795066
\(775\) 0 0
\(776\) −2.36669 −0.0849593
\(777\) 4.21110 0.151072
\(778\) 66.3583 2.37906
\(779\) 0 0
\(780\) 0 0
\(781\) 9.00000 0.322045
\(782\) 65.8722 2.35558
\(783\) 7.60555 0.271800
\(784\) 0.302776 0.0108134
\(785\) 0 0
\(786\) −17.0278 −0.607360
\(787\) −28.2389 −1.00661 −0.503303 0.864110i \(-0.667882\pi\)
−0.503303 + 0.864110i \(0.667882\pi\)
\(788\) −55.5416 −1.97859
\(789\) −5.78890 −0.206090
\(790\) 0 0
\(791\) −10.8167 −0.384596
\(792\) 9.00000 0.319801
\(793\) −3.15559 −0.112058
\(794\) 4.60555 0.163445
\(795\) 0 0
\(796\) 26.4222 0.936510
\(797\) 29.4500 1.04317 0.521586 0.853199i \(-0.325341\pi\)
0.521586 + 0.853199i \(0.325341\pi\)
\(798\) −15.2111 −0.538467
\(799\) 48.8444 1.72799
\(800\) 0 0
\(801\) 7.81665 0.276188
\(802\) 38.7250 1.36743
\(803\) 1.81665 0.0641083
\(804\) 51.5416 1.81773
\(805\) 0 0
\(806\) 43.2666 1.52400
\(807\) 3.21110 0.113036
\(808\) −49.8167 −1.75254
\(809\) 32.4500 1.14088 0.570440 0.821339i \(-0.306773\pi\)
0.570440 + 0.821339i \(0.306773\pi\)
\(810\) 0 0
\(811\) 14.8444 0.521258 0.260629 0.965439i \(-0.416070\pi\)
0.260629 + 0.965439i \(0.416070\pi\)
\(812\) 25.1194 0.881519
\(813\) 26.6056 0.933097
\(814\) 29.0917 1.01966
\(815\) 0 0
\(816\) 1.39445 0.0488155
\(817\) −63.4500 −2.21983
\(818\) 27.2111 0.951414
\(819\) −2.60555 −0.0910453
\(820\) 0 0
\(821\) 49.2666 1.71942 0.859708 0.510785i \(-0.170645\pi\)
0.859708 + 0.510785i \(0.170645\pi\)
\(822\) 7.39445 0.257911
\(823\) −24.8167 −0.865054 −0.432527 0.901621i \(-0.642378\pi\)
−0.432527 + 0.901621i \(0.642378\pi\)
\(824\) 11.4500 0.398878
\(825\) 0 0
\(826\) −24.4222 −0.849757
\(827\) 30.6333 1.06522 0.532612 0.846359i \(-0.321210\pi\)
0.532612 + 0.846359i \(0.321210\pi\)
\(828\) 20.5139 0.712907
\(829\) −0.238859 −0.00829591 −0.00414796 0.999991i \(-0.501320\pi\)
−0.00414796 + 0.999991i \(0.501320\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) −33.3944 −1.15774
\(833\) −4.60555 −0.159573
\(834\) 43.8167 1.51725
\(835\) 0 0
\(836\) −65.4500 −2.26363
\(837\) 7.21110 0.249252
\(838\) −13.8167 −0.477288
\(839\) −22.1833 −0.765854 −0.382927 0.923779i \(-0.625084\pi\)
−0.382927 + 0.923779i \(0.625084\pi\)
\(840\) 0 0
\(841\) 28.8444 0.994635
\(842\) 23.5139 0.810342
\(843\) 1.18335 0.0407566
\(844\) 87.2666 3.00384
\(845\) 0 0
\(846\) 24.4222 0.839653
\(847\) 2.00000 0.0687208
\(848\) −0.972244 −0.0333870
\(849\) −13.6333 −0.467894
\(850\) 0 0
\(851\) 26.1556 0.896602
\(852\) 9.90833 0.339454
\(853\) 6.78890 0.232447 0.116224 0.993223i \(-0.462921\pi\)
0.116224 + 0.993223i \(0.462921\pi\)
\(854\) −2.78890 −0.0954341
\(855\) 0 0
\(856\) 0 0
\(857\) −27.6333 −0.943936 −0.471968 0.881616i \(-0.656456\pi\)
−0.471968 + 0.881616i \(0.656456\pi\)
\(858\) −18.0000 −0.614510
\(859\) −49.6333 −1.69347 −0.846733 0.532018i \(-0.821434\pi\)
−0.846733 + 0.532018i \(0.821434\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −12.8444 −0.437482
\(863\) −5.36669 −0.182684 −0.0913422 0.995820i \(-0.529116\pi\)
−0.0913422 + 0.995820i \(0.529116\pi\)
\(864\) −5.30278 −0.180404
\(865\) 0 0
\(866\) −85.6888 −2.91182
\(867\) −4.21110 −0.143017
\(868\) 23.8167 0.808390
\(869\) 44.4500 1.50786
\(870\) 0 0
\(871\) −40.6611 −1.37775
\(872\) −6.63331 −0.224632
\(873\) 0.788897 0.0267001
\(874\) −94.4777 −3.19576
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) 22.0000 0.742887 0.371444 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) −15.2111 −0.513350
\(879\) 10.6056 0.357716
\(880\) 0 0
\(881\) 33.6333 1.13313 0.566567 0.824015i \(-0.308271\pi\)
0.566567 + 0.824015i \(0.308271\pi\)
\(882\) −2.30278 −0.0775385
\(883\) −15.6056 −0.525169 −0.262584 0.964909i \(-0.584575\pi\)
−0.262584 + 0.964909i \(0.584575\pi\)
\(884\) −39.6333 −1.33301
\(885\) 0 0
\(886\) 36.0000 1.20944
\(887\) 14.7889 0.496563 0.248281 0.968688i \(-0.420134\pi\)
0.248281 + 0.968688i \(0.420134\pi\)
\(888\) 12.6333 0.423946
\(889\) 12.8167 0.429857
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) −50.8444 −1.70240
\(893\) −70.0555 −2.34432
\(894\) −37.7527 −1.26264
\(895\) 0 0
\(896\) −18.9083 −0.631683
\(897\) −16.1833 −0.540346
\(898\) 80.1749 2.67547
\(899\) 54.8444 1.82916
\(900\) 0 0
\(901\) 14.7889 0.492690
\(902\) 0 0
\(903\) −9.60555 −0.319653
\(904\) −32.4500 −1.07927
\(905\) 0 0
\(906\) 15.6972 0.521505
\(907\) 13.2111 0.438667 0.219334 0.975650i \(-0.429612\pi\)
0.219334 + 0.975650i \(0.429612\pi\)
\(908\) −74.6611 −2.47771
\(909\) 16.6056 0.550771
\(910\) 0 0
\(911\) 15.0000 0.496972 0.248486 0.968635i \(-0.420067\pi\)
0.248486 + 0.968635i \(0.420067\pi\)
\(912\) −2.00000 −0.0662266
\(913\) −9.63331 −0.318816
\(914\) −8.72498 −0.288597
\(915\) 0 0
\(916\) −23.8167 −0.786924
\(917\) 7.39445 0.244186
\(918\) −10.6056 −0.350035
\(919\) −17.6056 −0.580754 −0.290377 0.956912i \(-0.593781\pi\)
−0.290377 + 0.956912i \(0.593781\pi\)
\(920\) 0 0
\(921\) −8.60555 −0.283563
\(922\) 60.4222 1.98990
\(923\) −7.81665 −0.257288
\(924\) −9.90833 −0.325960
\(925\) 0 0
\(926\) 39.6333 1.30243
\(927\) −3.81665 −0.125355
\(928\) −40.3305 −1.32391
\(929\) −20.2389 −0.664015 −0.332008 0.943277i \(-0.607726\pi\)
−0.332008 + 0.943277i \(0.607726\pi\)
\(930\) 0 0
\(931\) 6.60555 0.216488
\(932\) 3.90833 0.128022
\(933\) −13.8167 −0.452337
\(934\) −0.972244 −0.0318128
\(935\) 0 0
\(936\) −7.81665 −0.255495
\(937\) 56.4777 1.84505 0.922523 0.385941i \(-0.126123\pi\)
0.922523 + 0.385941i \(0.126123\pi\)
\(938\) −35.9361 −1.17335
\(939\) −23.8167 −0.777227
\(940\) 0 0
\(941\) 6.00000 0.195594 0.0977972 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) −33.2111 −1.08208
\(943\) 0 0
\(944\) −3.21110 −0.104512
\(945\) 0 0
\(946\) −66.3583 −2.15749
\(947\) 0.844410 0.0274396 0.0137198 0.999906i \(-0.495633\pi\)
0.0137198 + 0.999906i \(0.495633\pi\)
\(948\) 48.9361 1.58937
\(949\) −1.57779 −0.0512174
\(950\) 0 0
\(951\) −16.3944 −0.531626
\(952\) −13.8167 −0.447800
\(953\) 13.6056 0.440727 0.220364 0.975418i \(-0.429276\pi\)
0.220364 + 0.975418i \(0.429276\pi\)
\(954\) 7.39445 0.239404
\(955\) 0 0
\(956\) 48.8444 1.57974
\(957\) −22.8167 −0.737558
\(958\) −79.6888 −2.57463
\(959\) −3.21110 −0.103692
\(960\) 0 0
\(961\) 21.0000 0.677419
\(962\) −25.2666 −0.814628
\(963\) 0 0
\(964\) 31.0278 0.999337
\(965\) 0 0
\(966\) −14.3028 −0.460184
\(967\) 40.8444 1.31347 0.656734 0.754122i \(-0.271937\pi\)
0.656734 + 0.754122i \(0.271937\pi\)
\(968\) 6.00000 0.192847
\(969\) 30.4222 0.977302
\(970\) 0 0
\(971\) 53.8722 1.72884 0.864420 0.502770i \(-0.167686\pi\)
0.864420 + 0.502770i \(0.167686\pi\)
\(972\) −3.30278 −0.105937
\(973\) −19.0278 −0.610002
\(974\) −47.9361 −1.53597
\(975\) 0 0
\(976\) −0.366692 −0.0117375
\(977\) 2.02776 0.0648737 0.0324368 0.999474i \(-0.489673\pi\)
0.0324368 + 0.999474i \(0.489673\pi\)
\(978\) −39.6333 −1.26733
\(979\) −23.4500 −0.749464
\(980\) 0 0
\(981\) 2.21110 0.0705951
\(982\) 6.90833 0.220454
\(983\) −40.0555 −1.27757 −0.638786 0.769384i \(-0.720563\pi\)
−0.638786 + 0.769384i \(0.720563\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −80.6611 −2.56877
\(987\) −10.6056 −0.337578
\(988\) 56.8444 1.80846
\(989\) −59.6611 −1.89711
\(990\) 0 0
\(991\) 46.0278 1.46212 0.731060 0.682313i \(-0.239026\pi\)
0.731060 + 0.682313i \(0.239026\pi\)
\(992\) −38.2389 −1.21408
\(993\) 21.2389 0.673995
\(994\) −6.90833 −0.219119
\(995\) 0 0
\(996\) −10.6056 −0.336050
\(997\) 15.4500 0.489305 0.244653 0.969611i \(-0.421326\pi\)
0.244653 + 0.969611i \(0.421326\pi\)
\(998\) 64.4777 2.04101
\(999\) −4.21110 −0.133233
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 525.2.a.f.1.1 2
3.2 odd 2 1575.2.a.t.1.2 2
4.3 odd 2 8400.2.a.df.1.2 2
5.2 odd 4 525.2.d.d.274.1 4
5.3 odd 4 525.2.d.d.274.4 4
5.4 even 2 525.2.a.h.1.2 yes 2
7.6 odd 2 3675.2.a.w.1.1 2
15.2 even 4 1575.2.d.g.1324.4 4
15.8 even 4 1575.2.d.g.1324.1 4
15.14 odd 2 1575.2.a.o.1.1 2
20.19 odd 2 8400.2.a.cw.1.1 2
35.34 odd 2 3675.2.a.bb.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
525.2.a.f.1.1 2 1.1 even 1 trivial
525.2.a.h.1.2 yes 2 5.4 even 2
525.2.d.d.274.1 4 5.2 odd 4
525.2.d.d.274.4 4 5.3 odd 4
1575.2.a.o.1.1 2 15.14 odd 2
1575.2.a.t.1.2 2 3.2 odd 2
1575.2.d.g.1324.1 4 15.8 even 4
1575.2.d.g.1324.4 4 15.2 even 4
3675.2.a.w.1.1 2 7.6 odd 2
3675.2.a.bb.1.2 2 35.34 odd 2
8400.2.a.cw.1.1 2 20.19 odd 2
8400.2.a.df.1.2 2 4.3 odd 2