Properties

Label 3675.2.a.bn.1.4
Level $3675$
Weight $2$
Character 3675.1
Self dual yes
Analytic conductor $29.345$
Analytic rank $1$
Dimension $4$
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] = [3675,2,Mod(1,3675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3675, 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("3675.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3675.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: \(29.3450227428\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.11344.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.78165\) of defining polynomial
Character \(\chi\) \(=\) 3675.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.78165 q^{2} -1.00000 q^{3} +1.17429 q^{4} -1.78165 q^{6} -1.47113 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.78165 q^{2} -1.00000 q^{3} +1.17429 q^{4} -1.78165 q^{6} -1.47113 q^{8} +1.00000 q^{9} -2.07850 q^{11} -1.17429 q^{12} +3.13023 q^{13} -4.96962 q^{16} -2.13023 q^{17} +1.78165 q^{18} +7.73760 q^{19} -3.70316 q^{22} -5.53655 q^{23} +1.47113 q^{24} +5.57699 q^{26} -1.00000 q^{27} -4.01368 q^{29} -2.91188 q^{31} -5.91188 q^{32} +2.07850 q^{33} -3.79533 q^{34} +1.17429 q^{36} -3.51519 q^{37} +13.7857 q^{38} -3.13023 q^{39} -7.99038 q^{41} +4.99038 q^{43} -2.44075 q^{44} -9.86421 q^{46} +2.44075 q^{47} +4.96962 q^{48} +2.13023 q^{51} +3.67580 q^{52} -9.91188 q^{53} -1.78165 q^{54} -7.73760 q^{57} -7.15099 q^{58} -2.95594 q^{59} +10.8946 q^{61} -5.18797 q^{62} -0.593684 q^{64} +3.70316 q^{66} -3.83939 q^{67} -2.50151 q^{68} +5.53655 q^{69} -15.0248 q^{71} -1.47113 q^{72} -8.55369 q^{73} -6.26285 q^{74} +9.08617 q^{76} -5.57699 q^{78} +8.11354 q^{79} +1.00000 q^{81} -14.2361 q^{82} -8.75128 q^{83} +8.89113 q^{86} +4.01368 q^{87} +3.05774 q^{88} -0.618661 q^{89} -6.50151 q^{92} +2.91188 q^{93} +4.34858 q^{94} +5.91188 q^{96} -0.296842 q^{97} -2.07850 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{6} - 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{6} - 6 q^{8} + 4 q^{9} - 4 q^{12} + 2 q^{13} + 2 q^{17} - 2 q^{18} + 12 q^{19} - 14 q^{22} - 10 q^{23} + 6 q^{24} - 6 q^{26} - 4 q^{27} - 6 q^{29} + 8 q^{31} - 4 q^{32} + 4 q^{34} + 4 q^{36} - 24 q^{37} + 8 q^{38} - 2 q^{39} - 4 q^{41} - 8 q^{43} + 10 q^{44} + 16 q^{46} - 10 q^{47} - 2 q^{51} + 34 q^{52} - 20 q^{53} + 2 q^{54} - 12 q^{57} - 10 q^{58} - 2 q^{59} + 8 q^{61} - 10 q^{62} - 4 q^{64} + 14 q^{66} - 6 q^{67} - 30 q^{68} + 10 q^{69} - 14 q^{71} - 6 q^{72} + 12 q^{73} + 20 q^{74} + 16 q^{76} + 6 q^{78} - 8 q^{79} + 4 q^{81} - 18 q^{82} - 6 q^{83} + 24 q^{86} + 6 q^{87} + 12 q^{88} - 8 q^{89} - 46 q^{92} - 8 q^{93} + 16 q^{94} + 4 q^{96} - 2 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.78165 1.25982 0.629910 0.776668i \(-0.283092\pi\)
0.629910 + 0.776668i \(0.283092\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.17429 0.587145
\(5\) 0 0
\(6\) −1.78165 −0.727357
\(7\) 0 0
\(8\) −1.47113 −0.520123
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.07850 −0.626690 −0.313345 0.949639i \(-0.601450\pi\)
−0.313345 + 0.949639i \(0.601450\pi\)
\(12\) −1.17429 −0.338988
\(13\) 3.13023 0.868170 0.434085 0.900872i \(-0.357072\pi\)
0.434085 + 0.900872i \(0.357072\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.96962 −1.24241
\(17\) −2.13023 −0.516657 −0.258329 0.966057i \(-0.583172\pi\)
−0.258329 + 0.966057i \(0.583172\pi\)
\(18\) 1.78165 0.419940
\(19\) 7.73760 1.77513 0.887563 0.460686i \(-0.152397\pi\)
0.887563 + 0.460686i \(0.152397\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −3.70316 −0.789516
\(23\) −5.53655 −1.15445 −0.577225 0.816585i \(-0.695864\pi\)
−0.577225 + 0.816585i \(0.695864\pi\)
\(24\) 1.47113 0.300293
\(25\) 0 0
\(26\) 5.57699 1.09374
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −4.01368 −0.745322 −0.372661 0.927968i \(-0.621555\pi\)
−0.372661 + 0.927968i \(0.621555\pi\)
\(30\) 0 0
\(31\) −2.91188 −0.522990 −0.261495 0.965205i \(-0.584215\pi\)
−0.261495 + 0.965205i \(0.584215\pi\)
\(32\) −5.91188 −1.04508
\(33\) 2.07850 0.361820
\(34\) −3.79533 −0.650894
\(35\) 0 0
\(36\) 1.17429 0.195715
\(37\) −3.51519 −0.577893 −0.288947 0.957345i \(-0.593305\pi\)
−0.288947 + 0.957345i \(0.593305\pi\)
\(38\) 13.7857 2.23634
\(39\) −3.13023 −0.501238
\(40\) 0 0
\(41\) −7.99038 −1.24789 −0.623944 0.781469i \(-0.714471\pi\)
−0.623944 + 0.781469i \(0.714471\pi\)
\(42\) 0 0
\(43\) 4.99038 0.761026 0.380513 0.924776i \(-0.375747\pi\)
0.380513 + 0.924776i \(0.375747\pi\)
\(44\) −2.44075 −0.367958
\(45\) 0 0
\(46\) −9.86421 −1.45440
\(47\) 2.44075 0.356021 0.178010 0.984029i \(-0.443034\pi\)
0.178010 + 0.984029i \(0.443034\pi\)
\(48\) 4.96962 0.717303
\(49\) 0 0
\(50\) 0 0
\(51\) 2.13023 0.298292
\(52\) 3.67580 0.509741
\(53\) −9.91188 −1.36150 −0.680751 0.732515i \(-0.738346\pi\)
−0.680751 + 0.732515i \(0.738346\pi\)
\(54\) −1.78165 −0.242452
\(55\) 0 0
\(56\) 0 0
\(57\) −7.73760 −1.02487
\(58\) −7.15099 −0.938971
\(59\) −2.95594 −0.384831 −0.192415 0.981314i \(-0.561632\pi\)
−0.192415 + 0.981314i \(0.561632\pi\)
\(60\) 0 0
\(61\) 10.8946 1.39491 0.697454 0.716629i \(-0.254316\pi\)
0.697454 + 0.716629i \(0.254316\pi\)
\(62\) −5.18797 −0.658873
\(63\) 0 0
\(64\) −0.593684 −0.0742104
\(65\) 0 0
\(66\) 3.70316 0.455827
\(67\) −3.83939 −0.469056 −0.234528 0.972109i \(-0.575355\pi\)
−0.234528 + 0.972109i \(0.575355\pi\)
\(68\) −2.50151 −0.303352
\(69\) 5.53655 0.666522
\(70\) 0 0
\(71\) −15.0248 −1.78312 −0.891559 0.452905i \(-0.850388\pi\)
−0.891559 + 0.452905i \(0.850388\pi\)
\(72\) −1.47113 −0.173374
\(73\) −8.55369 −1.00113 −0.500567 0.865698i \(-0.666875\pi\)
−0.500567 + 0.865698i \(0.666875\pi\)
\(74\) −6.26285 −0.728041
\(75\) 0 0
\(76\) 9.08617 1.04226
\(77\) 0 0
\(78\) −5.57699 −0.631470
\(79\) 8.11354 0.912844 0.456422 0.889763i \(-0.349131\pi\)
0.456422 + 0.889763i \(0.349131\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −14.2361 −1.57211
\(83\) −8.75128 −0.960577 −0.480289 0.877110i \(-0.659468\pi\)
−0.480289 + 0.877110i \(0.659468\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.89113 0.958755
\(87\) 4.01368 0.430312
\(88\) 3.05774 0.325956
\(89\) −0.618661 −0.0655779 −0.0327890 0.999462i \(-0.510439\pi\)
−0.0327890 + 0.999462i \(0.510439\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −6.50151 −0.677829
\(93\) 2.91188 0.301948
\(94\) 4.34858 0.448522
\(95\) 0 0
\(96\) 5.91188 0.603379
\(97\) −0.296842 −0.0301397 −0.0150699 0.999886i \(-0.504797\pi\)
−0.0150699 + 0.999886i \(0.504797\pi\)
\(98\) 0 0
\(99\) −2.07850 −0.208897
\(100\) 0 0
\(101\) −8.25879 −0.821780 −0.410890 0.911685i \(-0.634782\pi\)
−0.410890 + 0.911685i \(0.634782\pi\)
\(102\) 3.79533 0.375794
\(103\) −17.0866 −1.68359 −0.841797 0.539794i \(-0.818502\pi\)
−0.841797 + 0.539794i \(0.818502\pi\)
\(104\) −4.60498 −0.451555
\(105\) 0 0
\(106\) −17.6595 −1.71525
\(107\) −6.31052 −0.610061 −0.305031 0.952343i \(-0.598667\pi\)
−0.305031 + 0.952343i \(0.598667\pi\)
\(108\) −1.17429 −0.112996
\(109\) 2.44676 0.234357 0.117178 0.993111i \(-0.462615\pi\)
0.117178 + 0.993111i \(0.462615\pi\)
\(110\) 0 0
\(111\) 3.51519 0.333647
\(112\) 0 0
\(113\) −10.1570 −0.955489 −0.477745 0.878499i \(-0.658546\pi\)
−0.477745 + 0.878499i \(0.658546\pi\)
\(114\) −13.7857 −1.29115
\(115\) 0 0
\(116\) −4.71322 −0.437612
\(117\) 3.13023 0.289390
\(118\) −5.26647 −0.484817
\(119\) 0 0
\(120\) 0 0
\(121\) −6.67986 −0.607260
\(122\) 19.4104 1.75733
\(123\) 7.99038 0.720468
\(124\) −3.41939 −0.307071
\(125\) 0 0
\(126\) 0 0
\(127\) −8.86977 −0.787065 −0.393532 0.919311i \(-0.628747\pi\)
−0.393532 + 0.919311i \(0.628747\pi\)
\(128\) 10.7660 0.951592
\(129\) −4.99038 −0.439378
\(130\) 0 0
\(131\) −5.34150 −0.466689 −0.233345 0.972394i \(-0.574967\pi\)
−0.233345 + 0.972394i \(0.574967\pi\)
\(132\) 2.44075 0.212440
\(133\) 0 0
\(134\) −6.84047 −0.590926
\(135\) 0 0
\(136\) 3.13385 0.268725
\(137\) 13.1039 1.11954 0.559772 0.828647i \(-0.310889\pi\)
0.559772 + 0.828647i \(0.310889\pi\)
\(138\) 9.86421 0.839697
\(139\) −0.243164 −0.0206249 −0.0103125 0.999947i \(-0.503283\pi\)
−0.0103125 + 0.999947i \(0.503283\pi\)
\(140\) 0 0
\(141\) −2.44075 −0.205549
\(142\) −26.7690 −2.24641
\(143\) −6.50617 −0.544073
\(144\) −4.96962 −0.414135
\(145\) 0 0
\(146\) −15.2397 −1.26125
\(147\) 0 0
\(148\) −4.12785 −0.339307
\(149\) 2.33728 0.191478 0.0957388 0.995406i \(-0.469479\pi\)
0.0957388 + 0.995406i \(0.469479\pi\)
\(150\) 0 0
\(151\) −11.1135 −0.904407 −0.452203 0.891915i \(-0.649362\pi\)
−0.452203 + 0.891915i \(0.649362\pi\)
\(152\) −11.3830 −0.923284
\(153\) −2.13023 −0.172219
\(154\) 0 0
\(155\) 0 0
\(156\) −3.67580 −0.294299
\(157\) 11.3182 0.903291 0.451645 0.892198i \(-0.350837\pi\)
0.451645 + 0.892198i \(0.350837\pi\)
\(158\) 14.4555 1.15002
\(159\) 9.91188 0.786064
\(160\) 0 0
\(161\) 0 0
\(162\) 1.78165 0.139980
\(163\) 2.10347 0.164757 0.0823783 0.996601i \(-0.473748\pi\)
0.0823783 + 0.996601i \(0.473748\pi\)
\(164\) −9.38302 −0.732690
\(165\) 0 0
\(166\) −15.5917 −1.21015
\(167\) −3.58600 −0.277493 −0.138747 0.990328i \(-0.544307\pi\)
−0.138747 + 0.990328i \(0.544307\pi\)
\(168\) 0 0
\(169\) −3.20165 −0.246281
\(170\) 0 0
\(171\) 7.73760 0.591709
\(172\) 5.86015 0.446832
\(173\) 15.5400 1.18148 0.590742 0.806860i \(-0.298835\pi\)
0.590742 + 0.806860i \(0.298835\pi\)
\(174\) 7.15099 0.542115
\(175\) 0 0
\(176\) 10.3293 0.778603
\(177\) 2.95594 0.222182
\(178\) −1.10224 −0.0826163
\(179\) 15.7357 1.17614 0.588069 0.808811i \(-0.299888\pi\)
0.588069 + 0.808811i \(0.299888\pi\)
\(180\) 0 0
\(181\) 4.98692 0.370675 0.185337 0.982675i \(-0.440662\pi\)
0.185337 + 0.982675i \(0.440662\pi\)
\(182\) 0 0
\(183\) −10.8946 −0.805351
\(184\) 8.14499 0.600456
\(185\) 0 0
\(186\) 5.18797 0.380400
\(187\) 4.42768 0.323784
\(188\) 2.86615 0.209036
\(189\) 0 0
\(190\) 0 0
\(191\) 11.8084 0.854427 0.427213 0.904151i \(-0.359495\pi\)
0.427213 + 0.904151i \(0.359495\pi\)
\(192\) 0.593684 0.0428454
\(193\) −24.7380 −1.78068 −0.890342 0.455293i \(-0.849534\pi\)
−0.890342 + 0.455293i \(0.849534\pi\)
\(194\) −0.528869 −0.0379706
\(195\) 0 0
\(196\) 0 0
\(197\) −19.5526 −1.39307 −0.696533 0.717525i \(-0.745275\pi\)
−0.696533 + 0.717525i \(0.745275\pi\)
\(198\) −3.70316 −0.263172
\(199\) 22.2401 1.57656 0.788281 0.615315i \(-0.210971\pi\)
0.788281 + 0.615315i \(0.210971\pi\)
\(200\) 0 0
\(201\) 3.83939 0.270810
\(202\) −14.7143 −1.03529
\(203\) 0 0
\(204\) 2.50151 0.175141
\(205\) 0 0
\(206\) −30.4424 −2.12102
\(207\) −5.53655 −0.384817
\(208\) −15.5561 −1.07862
\(209\) −16.0826 −1.11245
\(210\) 0 0
\(211\) 23.6191 1.62601 0.813003 0.582259i \(-0.197831\pi\)
0.813003 + 0.582259i \(0.197831\pi\)
\(212\) −11.6394 −0.799398
\(213\) 15.0248 1.02948
\(214\) −11.2432 −0.768567
\(215\) 0 0
\(216\) 1.47113 0.100098
\(217\) 0 0
\(218\) 4.35927 0.295247
\(219\) 8.55369 0.578005
\(220\) 0 0
\(221\) −6.66812 −0.448546
\(222\) 6.26285 0.420335
\(223\) −25.1420 −1.68363 −0.841815 0.539765i \(-0.818513\pi\)
−0.841815 + 0.539765i \(0.818513\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −18.0962 −1.20374
\(227\) 23.6702 1.57105 0.785524 0.618831i \(-0.212393\pi\)
0.785524 + 0.618831i \(0.212393\pi\)
\(228\) −9.08617 −0.601747
\(229\) −0.406316 −0.0268501 −0.0134251 0.999910i \(-0.504273\pi\)
−0.0134251 + 0.999910i \(0.504273\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 5.90465 0.387659
\(233\) 7.16527 0.469413 0.234706 0.972066i \(-0.424587\pi\)
0.234706 + 0.972066i \(0.424587\pi\)
\(234\) 5.57699 0.364579
\(235\) 0 0
\(236\) −3.47113 −0.225951
\(237\) −8.11354 −0.527031
\(238\) 0 0
\(239\) 10.0922 0.652809 0.326404 0.945230i \(-0.394163\pi\)
0.326404 + 0.945230i \(0.394163\pi\)
\(240\) 0 0
\(241\) 4.60676 0.296748 0.148374 0.988931i \(-0.452596\pi\)
0.148374 + 0.988931i \(0.452596\pi\)
\(242\) −11.9012 −0.765038
\(243\) −1.00000 −0.0641500
\(244\) 12.7934 0.819013
\(245\) 0 0
\(246\) 14.2361 0.907660
\(247\) 24.2205 1.54111
\(248\) 4.28376 0.272019
\(249\) 8.75128 0.554590
\(250\) 0 0
\(251\) 0.311597 0.0196678 0.00983390 0.999952i \(-0.496870\pi\)
0.00983390 + 0.999952i \(0.496870\pi\)
\(252\) 0 0
\(253\) 11.5077 0.723482
\(254\) −15.8029 −0.991559
\(255\) 0 0
\(256\) 20.3687 1.27304
\(257\) −2.50211 −0.156077 −0.0780387 0.996950i \(-0.524866\pi\)
−0.0780387 + 0.996950i \(0.524866\pi\)
\(258\) −8.89113 −0.553537
\(259\) 0 0
\(260\) 0 0
\(261\) −4.01368 −0.248441
\(262\) −9.51671 −0.587944
\(263\) 4.62511 0.285196 0.142598 0.989781i \(-0.454454\pi\)
0.142598 + 0.989781i \(0.454454\pi\)
\(264\) −3.05774 −0.188191
\(265\) 0 0
\(266\) 0 0
\(267\) 0.618661 0.0378614
\(268\) −4.50856 −0.275404
\(269\) −12.0233 −0.733074 −0.366537 0.930404i \(-0.619457\pi\)
−0.366537 + 0.930404i \(0.619457\pi\)
\(270\) 0 0
\(271\) 5.46935 0.332239 0.166120 0.986106i \(-0.446876\pi\)
0.166120 + 0.986106i \(0.446876\pi\)
\(272\) 10.5864 0.641898
\(273\) 0 0
\(274\) 23.3466 1.41042
\(275\) 0 0
\(276\) 6.50151 0.391345
\(277\) −29.8099 −1.79111 −0.895553 0.444956i \(-0.853219\pi\)
−0.895553 + 0.444956i \(0.853219\pi\)
\(278\) −0.433235 −0.0259837
\(279\) −2.91188 −0.174330
\(280\) 0 0
\(281\) 7.78511 0.464421 0.232210 0.972666i \(-0.425404\pi\)
0.232210 + 0.972666i \(0.425404\pi\)
\(282\) −4.34858 −0.258954
\(283\) −2.61444 −0.155412 −0.0777062 0.996976i \(-0.524760\pi\)
−0.0777062 + 0.996976i \(0.524760\pi\)
\(284\) −17.6435 −1.04695
\(285\) 0 0
\(286\) −11.5917 −0.685434
\(287\) 0 0
\(288\) −5.91188 −0.348361
\(289\) −12.4621 −0.733066
\(290\) 0 0
\(291\) 0.296842 0.0174012
\(292\) −10.0445 −0.587810
\(293\) 12.7559 0.745210 0.372605 0.927990i \(-0.378465\pi\)
0.372605 + 0.927990i \(0.378465\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 5.17130 0.300576
\(297\) 2.07850 0.120607
\(298\) 4.16423 0.241227
\(299\) −17.3307 −1.00226
\(300\) 0 0
\(301\) 0 0
\(302\) −19.8005 −1.13939
\(303\) 8.25879 0.474455
\(304\) −38.4529 −2.20543
\(305\) 0 0
\(306\) −3.79533 −0.216965
\(307\) 13.1919 0.752900 0.376450 0.926437i \(-0.377145\pi\)
0.376450 + 0.926437i \(0.377145\pi\)
\(308\) 0 0
\(309\) 17.0866 0.972024
\(310\) 0 0
\(311\) 14.2823 0.809873 0.404937 0.914345i \(-0.367294\pi\)
0.404937 + 0.914345i \(0.367294\pi\)
\(312\) 4.60498 0.260706
\(313\) −4.77143 −0.269697 −0.134849 0.990866i \(-0.543055\pi\)
−0.134849 + 0.990866i \(0.543055\pi\)
\(314\) 20.1651 1.13798
\(315\) 0 0
\(316\) 9.52764 0.535971
\(317\) −18.8048 −1.05618 −0.528091 0.849188i \(-0.677092\pi\)
−0.528091 + 0.849188i \(0.677092\pi\)
\(318\) 17.6595 0.990298
\(319\) 8.34242 0.467086
\(320\) 0 0
\(321\) 6.31052 0.352219
\(322\) 0 0
\(323\) −16.4829 −0.917131
\(324\) 1.17429 0.0652383
\(325\) 0 0
\(326\) 3.74766 0.207564
\(327\) −2.44676 −0.135306
\(328\) 11.7549 0.649055
\(329\) 0 0
\(330\) 0 0
\(331\) 17.7742 0.976956 0.488478 0.872576i \(-0.337552\pi\)
0.488478 + 0.872576i \(0.337552\pi\)
\(332\) −10.2765 −0.563998
\(333\) −3.51519 −0.192631
\(334\) −6.38902 −0.349592
\(335\) 0 0
\(336\) 0 0
\(337\) 3.86675 0.210635 0.105318 0.994439i \(-0.466414\pi\)
0.105318 + 0.994439i \(0.466414\pi\)
\(338\) −5.70423 −0.310269
\(339\) 10.1570 0.551652
\(340\) 0 0
\(341\) 6.05234 0.327753
\(342\) 13.7857 0.745446
\(343\) 0 0
\(344\) −7.34150 −0.395827
\(345\) 0 0
\(346\) 27.6869 1.48846
\(347\) 10.2725 0.551455 0.275727 0.961236i \(-0.411081\pi\)
0.275727 + 0.961236i \(0.411081\pi\)
\(348\) 4.71322 0.252655
\(349\) −23.6180 −1.26424 −0.632122 0.774869i \(-0.717816\pi\)
−0.632122 + 0.774869i \(0.717816\pi\)
\(350\) 0 0
\(351\) −3.13023 −0.167079
\(352\) 12.2878 0.654943
\(353\) −5.77759 −0.307510 −0.153755 0.988109i \(-0.549137\pi\)
−0.153755 + 0.988109i \(0.549137\pi\)
\(354\) 5.26647 0.279909
\(355\) 0 0
\(356\) −0.726487 −0.0385037
\(357\) 0 0
\(358\) 28.0355 1.48172
\(359\) 31.3054 1.65224 0.826118 0.563497i \(-0.190544\pi\)
0.826118 + 0.563497i \(0.190544\pi\)
\(360\) 0 0
\(361\) 40.8704 2.15107
\(362\) 8.88497 0.466983
\(363\) 6.67986 0.350602
\(364\) 0 0
\(365\) 0 0
\(366\) −19.4104 −1.01460
\(367\) −16.5519 −0.864002 −0.432001 0.901873i \(-0.642192\pi\)
−0.432001 + 0.901873i \(0.642192\pi\)
\(368\) 27.5146 1.43430
\(369\) −7.99038 −0.415963
\(370\) 0 0
\(371\) 0 0
\(372\) 3.41939 0.177287
\(373\) −32.1426 −1.66428 −0.832140 0.554566i \(-0.812884\pi\)
−0.832140 + 0.554566i \(0.812884\pi\)
\(374\) 7.88858 0.407909
\(375\) 0 0
\(376\) −3.59067 −0.185175
\(377\) −12.5638 −0.647066
\(378\) 0 0
\(379\) −4.98800 −0.256216 −0.128108 0.991760i \(-0.540890\pi\)
−0.128108 + 0.991760i \(0.540890\pi\)
\(380\) 0 0
\(381\) 8.86977 0.454412
\(382\) 21.0385 1.07642
\(383\) 26.4568 1.35188 0.675940 0.736957i \(-0.263738\pi\)
0.675940 + 0.736957i \(0.263738\pi\)
\(384\) −10.7660 −0.549402
\(385\) 0 0
\(386\) −44.0746 −2.24334
\(387\) 4.99038 0.253675
\(388\) −0.348578 −0.0176964
\(389\) −12.7716 −0.647545 −0.323773 0.946135i \(-0.604951\pi\)
−0.323773 + 0.946135i \(0.604951\pi\)
\(390\) 0 0
\(391\) 11.7941 0.596455
\(392\) 0 0
\(393\) 5.34150 0.269443
\(394\) −34.8360 −1.75501
\(395\) 0 0
\(396\) −2.44075 −0.122653
\(397\) −15.1212 −0.758912 −0.379456 0.925210i \(-0.623889\pi\)
−0.379456 + 0.925210i \(0.623889\pi\)
\(398\) 39.6242 1.98618
\(399\) 0 0
\(400\) 0 0
\(401\) 12.3606 0.617258 0.308629 0.951182i \(-0.400130\pi\)
0.308629 + 0.951182i \(0.400130\pi\)
\(402\) 6.84047 0.341171
\(403\) −9.11487 −0.454044
\(404\) −9.69820 −0.482504
\(405\) 0 0
\(406\) 0 0
\(407\) 7.30630 0.362160
\(408\) −3.13385 −0.155149
\(409\) 10.5287 0.520611 0.260306 0.965526i \(-0.416177\pi\)
0.260306 + 0.965526i \(0.416177\pi\)
\(410\) 0 0
\(411\) −13.1039 −0.646369
\(412\) −20.0646 −0.988513
\(413\) 0 0
\(414\) −9.86421 −0.484799
\(415\) 0 0
\(416\) −18.5056 −0.907310
\(417\) 0.243164 0.0119078
\(418\) −28.6535 −1.40149
\(419\) −13.3110 −0.650283 −0.325142 0.945665i \(-0.605412\pi\)
−0.325142 + 0.945665i \(0.605412\pi\)
\(420\) 0 0
\(421\) 19.1520 0.933413 0.466707 0.884412i \(-0.345440\pi\)
0.466707 + 0.884412i \(0.345440\pi\)
\(422\) 42.0811 2.04847
\(423\) 2.44075 0.118674
\(424\) 14.5817 0.708149
\(425\) 0 0
\(426\) 26.7690 1.29696
\(427\) 0 0
\(428\) −7.41038 −0.358194
\(429\) 6.50617 0.314121
\(430\) 0 0
\(431\) 10.8964 0.524860 0.262430 0.964951i \(-0.415476\pi\)
0.262430 + 0.964951i \(0.415476\pi\)
\(432\) 4.96962 0.239101
\(433\) 19.4869 0.936482 0.468241 0.883601i \(-0.344888\pi\)
0.468241 + 0.883601i \(0.344888\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.87320 0.137601
\(437\) −42.8396 −2.04929
\(438\) 15.2397 0.728181
\(439\) −13.5310 −0.645799 −0.322899 0.946433i \(-0.604658\pi\)
−0.322899 + 0.946433i \(0.604658\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −11.8803 −0.565087
\(443\) −28.8663 −1.37148 −0.685740 0.727846i \(-0.740521\pi\)
−0.685740 + 0.727846i \(0.740521\pi\)
\(444\) 4.12785 0.195899
\(445\) 0 0
\(446\) −44.7943 −2.12107
\(447\) −2.33728 −0.110550
\(448\) 0 0
\(449\) 23.4298 1.10572 0.552860 0.833274i \(-0.313536\pi\)
0.552860 + 0.833274i \(0.313536\pi\)
\(450\) 0 0
\(451\) 16.6080 0.782039
\(452\) −11.9272 −0.561010
\(453\) 11.1135 0.522159
\(454\) 42.1722 1.97924
\(455\) 0 0
\(456\) 11.3830 0.533059
\(457\) −30.2876 −1.41679 −0.708396 0.705815i \(-0.750581\pi\)
−0.708396 + 0.705815i \(0.750581\pi\)
\(458\) −0.723915 −0.0338263
\(459\) 2.13023 0.0994307
\(460\) 0 0
\(461\) 7.02196 0.327045 0.163523 0.986540i \(-0.447714\pi\)
0.163523 + 0.986540i \(0.447714\pi\)
\(462\) 0 0
\(463\) −2.97324 −0.138178 −0.0690891 0.997610i \(-0.522009\pi\)
−0.0690891 + 0.997610i \(0.522009\pi\)
\(464\) 19.9465 0.925992
\(465\) 0 0
\(466\) 12.7660 0.591375
\(467\) 23.7549 1.09925 0.549623 0.835413i \(-0.314771\pi\)
0.549623 + 0.835413i \(0.314771\pi\)
\(468\) 3.67580 0.169914
\(469\) 0 0
\(470\) 0 0
\(471\) −11.3182 −0.521515
\(472\) 4.34858 0.200160
\(473\) −10.3725 −0.476927
\(474\) −14.4555 −0.663964
\(475\) 0 0
\(476\) 0 0
\(477\) −9.91188 −0.453834
\(478\) 17.9808 0.822421
\(479\) 6.54423 0.299013 0.149507 0.988761i \(-0.452231\pi\)
0.149507 + 0.988761i \(0.452231\pi\)
\(480\) 0 0
\(481\) −11.0034 −0.501710
\(482\) 8.20765 0.373848
\(483\) 0 0
\(484\) −7.84408 −0.356549
\(485\) 0 0
\(486\) −1.78165 −0.0808174
\(487\) −16.3269 −0.739844 −0.369922 0.929063i \(-0.620616\pi\)
−0.369922 + 0.929063i \(0.620616\pi\)
\(488\) −16.0274 −0.725525
\(489\) −2.10347 −0.0951223
\(490\) 0 0
\(491\) 24.9009 1.12376 0.561882 0.827218i \(-0.310078\pi\)
0.561882 + 0.827218i \(0.310078\pi\)
\(492\) 9.38302 0.423019
\(493\) 8.55007 0.385076
\(494\) 43.1525 1.94152
\(495\) 0 0
\(496\) 14.4710 0.649766
\(497\) 0 0
\(498\) 15.5917 0.698683
\(499\) −12.2039 −0.546323 −0.273161 0.961968i \(-0.588069\pi\)
−0.273161 + 0.961968i \(0.588069\pi\)
\(500\) 0 0
\(501\) 3.58600 0.160211
\(502\) 0.555157 0.0247779
\(503\) −27.8165 −1.24028 −0.620139 0.784492i \(-0.712924\pi\)
−0.620139 + 0.784492i \(0.712924\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 20.5027 0.911457
\(507\) 3.20165 0.142190
\(508\) −10.4157 −0.462121
\(509\) 27.8199 1.23309 0.616547 0.787318i \(-0.288531\pi\)
0.616547 + 0.787318i \(0.288531\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 14.7579 0.652214
\(513\) −7.73760 −0.341623
\(514\) −4.45789 −0.196629
\(515\) 0 0
\(516\) −5.86015 −0.257979
\(517\) −5.07310 −0.223115
\(518\) 0 0
\(519\) −15.5400 −0.682131
\(520\) 0 0
\(521\) −33.5026 −1.46777 −0.733887 0.679272i \(-0.762296\pi\)
−0.733887 + 0.679272i \(0.762296\pi\)
\(522\) −7.15099 −0.312990
\(523\) 12.7958 0.559522 0.279761 0.960070i \(-0.409745\pi\)
0.279761 + 0.960070i \(0.409745\pi\)
\(524\) −6.27247 −0.274014
\(525\) 0 0
\(526\) 8.24034 0.359296
\(527\) 6.20299 0.270206
\(528\) −10.3293 −0.449527
\(529\) 7.65336 0.332755
\(530\) 0 0
\(531\) −2.95594 −0.128277
\(532\) 0 0
\(533\) −25.0117 −1.08338
\(534\) 1.10224 0.0476986
\(535\) 0 0
\(536\) 5.64825 0.243967
\(537\) −15.7357 −0.679044
\(538\) −21.4214 −0.923540
\(539\) 0 0
\(540\) 0 0
\(541\) 25.2566 1.08587 0.542933 0.839776i \(-0.317314\pi\)
0.542933 + 0.839776i \(0.317314\pi\)
\(542\) 9.74448 0.418561
\(543\) −4.98692 −0.214009
\(544\) 12.5937 0.539950
\(545\) 0 0
\(546\) 0 0
\(547\) 38.8743 1.66214 0.831072 0.556165i \(-0.187728\pi\)
0.831072 + 0.556165i \(0.187728\pi\)
\(548\) 15.3878 0.657334
\(549\) 10.8946 0.464970
\(550\) 0 0
\(551\) −31.0562 −1.32304
\(552\) −8.14499 −0.346674
\(553\) 0 0
\(554\) −53.1110 −2.25647
\(555\) 0 0
\(556\) −0.285545 −0.0121098
\(557\) 1.36378 0.0577850 0.0288925 0.999583i \(-0.490802\pi\)
0.0288925 + 0.999583i \(0.490802\pi\)
\(558\) −5.18797 −0.219624
\(559\) 15.6210 0.660700
\(560\) 0 0
\(561\) −4.42768 −0.186937
\(562\) 13.8704 0.585086
\(563\) 10.5744 0.445660 0.222830 0.974857i \(-0.428471\pi\)
0.222830 + 0.974857i \(0.428471\pi\)
\(564\) −2.86615 −0.120687
\(565\) 0 0
\(566\) −4.65803 −0.195791
\(567\) 0 0
\(568\) 22.1035 0.927441
\(569\) 37.9865 1.59248 0.796238 0.604983i \(-0.206820\pi\)
0.796238 + 0.604983i \(0.206820\pi\)
\(570\) 0 0
\(571\) −15.6910 −0.656648 −0.328324 0.944565i \(-0.606484\pi\)
−0.328324 + 0.944565i \(0.606484\pi\)
\(572\) −7.64013 −0.319450
\(573\) −11.8084 −0.493304
\(574\) 0 0
\(575\) 0 0
\(576\) −0.593684 −0.0247368
\(577\) −3.44809 −0.143546 −0.0717730 0.997421i \(-0.522866\pi\)
−0.0717730 + 0.997421i \(0.522866\pi\)
\(578\) −22.2032 −0.923530
\(579\) 24.7380 1.02808
\(580\) 0 0
\(581\) 0 0
\(582\) 0.528869 0.0219223
\(583\) 20.6018 0.853240
\(584\) 12.5836 0.520713
\(585\) 0 0
\(586\) 22.7267 0.938830
\(587\) 10.5983 0.437441 0.218720 0.975788i \(-0.429812\pi\)
0.218720 + 0.975788i \(0.429812\pi\)
\(588\) 0 0
\(589\) −22.5310 −0.928373
\(590\) 0 0
\(591\) 19.5526 0.804287
\(592\) 17.4692 0.717978
\(593\) −0.234412 −0.00962614 −0.00481307 0.999988i \(-0.501532\pi\)
−0.00481307 + 0.999988i \(0.501532\pi\)
\(594\) 3.70316 0.151942
\(595\) 0 0
\(596\) 2.74464 0.112425
\(597\) −22.2401 −0.910229
\(598\) −30.8773 −1.26267
\(599\) −25.8290 −1.05535 −0.527673 0.849448i \(-0.676935\pi\)
−0.527673 + 0.849448i \(0.676935\pi\)
\(600\) 0 0
\(601\) 1.15592 0.0471508 0.0235754 0.999722i \(-0.492495\pi\)
0.0235754 + 0.999722i \(0.492495\pi\)
\(602\) 0 0
\(603\) −3.83939 −0.156352
\(604\) −13.0505 −0.531017
\(605\) 0 0
\(606\) 14.7143 0.597727
\(607\) 23.2111 0.942110 0.471055 0.882104i \(-0.343873\pi\)
0.471055 + 0.882104i \(0.343873\pi\)
\(608\) −45.7438 −1.85516
\(609\) 0 0
\(610\) 0 0
\(611\) 7.64013 0.309086
\(612\) −2.50151 −0.101117
\(613\) −6.33144 −0.255724 −0.127862 0.991792i \(-0.540812\pi\)
−0.127862 + 0.991792i \(0.540812\pi\)
\(614\) 23.5033 0.948518
\(615\) 0 0
\(616\) 0 0
\(617\) 25.9546 1.04489 0.522447 0.852672i \(-0.325019\pi\)
0.522447 + 0.852672i \(0.325019\pi\)
\(618\) 30.4424 1.22457
\(619\) 29.0962 1.16948 0.584738 0.811222i \(-0.301197\pi\)
0.584738 + 0.811222i \(0.301197\pi\)
\(620\) 0 0
\(621\) 5.53655 0.222174
\(622\) 25.4460 1.02029
\(623\) 0 0
\(624\) 15.5561 0.622741
\(625\) 0 0
\(626\) −8.50104 −0.339770
\(627\) 16.0826 0.642275
\(628\) 13.2908 0.530362
\(629\) 7.48816 0.298573
\(630\) 0 0
\(631\) −37.3609 −1.48731 −0.743657 0.668561i \(-0.766911\pi\)
−0.743657 + 0.668561i \(0.766911\pi\)
\(632\) −11.9361 −0.474791
\(633\) −23.6191 −0.938775
\(634\) −33.5036 −1.33060
\(635\) 0 0
\(636\) 11.6394 0.461533
\(637\) 0 0
\(638\) 14.8633 0.588443
\(639\) −15.0248 −0.594373
\(640\) 0 0
\(641\) −27.8535 −1.10015 −0.550074 0.835116i \(-0.685400\pi\)
−0.550074 + 0.835116i \(0.685400\pi\)
\(642\) 11.2432 0.443732
\(643\) 20.3104 0.800963 0.400481 0.916305i \(-0.368843\pi\)
0.400481 + 0.916305i \(0.368843\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −29.3668 −1.15542
\(647\) −22.4329 −0.881929 −0.440964 0.897525i \(-0.645364\pi\)
−0.440964 + 0.897525i \(0.645364\pi\)
\(648\) −1.47113 −0.0577915
\(649\) 6.14391 0.241170
\(650\) 0 0
\(651\) 0 0
\(652\) 2.47008 0.0967360
\(653\) 19.8051 0.775035 0.387517 0.921862i \(-0.373333\pi\)
0.387517 + 0.921862i \(0.373333\pi\)
\(654\) −4.35927 −0.170461
\(655\) 0 0
\(656\) 39.7092 1.55038
\(657\) −8.55369 −0.333711
\(658\) 0 0
\(659\) 38.3567 1.49416 0.747082 0.664731i \(-0.231454\pi\)
0.747082 + 0.664731i \(0.231454\pi\)
\(660\) 0 0
\(661\) −2.97492 −0.115711 −0.0578554 0.998325i \(-0.518426\pi\)
−0.0578554 + 0.998325i \(0.518426\pi\)
\(662\) 31.6674 1.23079
\(663\) 6.66812 0.258968
\(664\) 12.8743 0.499619
\(665\) 0 0
\(666\) −6.26285 −0.242680
\(667\) 22.2219 0.860437
\(668\) −4.21101 −0.162929
\(669\) 25.1420 0.972045
\(670\) 0 0
\(671\) −22.6443 −0.874175
\(672\) 0 0
\(673\) −22.8397 −0.880407 −0.440203 0.897898i \(-0.645094\pi\)
−0.440203 + 0.897898i \(0.645094\pi\)
\(674\) 6.88921 0.265363
\(675\) 0 0
\(676\) −3.75966 −0.144602
\(677\) −7.19609 −0.276568 −0.138284 0.990393i \(-0.544159\pi\)
−0.138284 + 0.990393i \(0.544159\pi\)
\(678\) 18.0962 0.694982
\(679\) 0 0
\(680\) 0 0
\(681\) −23.6702 −0.907045
\(682\) 10.7832 0.412909
\(683\) −4.55442 −0.174270 −0.0871351 0.996197i \(-0.527771\pi\)
−0.0871351 + 0.996197i \(0.527771\pi\)
\(684\) 9.08617 0.347419
\(685\) 0 0
\(686\) 0 0
\(687\) 0.406316 0.0155019
\(688\) −24.8003 −0.945503
\(689\) −31.0265 −1.18202
\(690\) 0 0
\(691\) −3.53896 −0.134628 −0.0673142 0.997732i \(-0.521443\pi\)
−0.0673142 + 0.997732i \(0.521443\pi\)
\(692\) 18.2485 0.693702
\(693\) 0 0
\(694\) 18.3020 0.694734
\(695\) 0 0
\(696\) −5.90465 −0.223815
\(697\) 17.0214 0.644730
\(698\) −42.0791 −1.59272
\(699\) −7.16527 −0.271015
\(700\) 0 0
\(701\) −16.8111 −0.634948 −0.317474 0.948267i \(-0.602835\pi\)
−0.317474 + 0.948267i \(0.602835\pi\)
\(702\) −5.57699 −0.210490
\(703\) −27.1991 −1.02583
\(704\) 1.23397 0.0465069
\(705\) 0 0
\(706\) −10.2937 −0.387407
\(707\) 0 0
\(708\) 3.47113 0.130453
\(709\) 31.8176 1.19494 0.597468 0.801893i \(-0.296174\pi\)
0.597468 + 0.801893i \(0.296174\pi\)
\(710\) 0 0
\(711\) 8.11354 0.304281
\(712\) 0.910131 0.0341086
\(713\) 16.1218 0.603766
\(714\) 0 0
\(715\) 0 0
\(716\) 18.4782 0.690563
\(717\) −10.0922 −0.376899
\(718\) 55.7754 2.08152
\(719\) −15.2807 −0.569876 −0.284938 0.958546i \(-0.591973\pi\)
−0.284938 + 0.958546i \(0.591973\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 72.8169 2.70996
\(723\) −4.60676 −0.171327
\(724\) 5.85609 0.217640
\(725\) 0 0
\(726\) 11.9012 0.441695
\(727\) 19.1829 0.711453 0.355726 0.934590i \(-0.384234\pi\)
0.355726 + 0.934590i \(0.384234\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −10.6307 −0.393189
\(732\) −12.7934 −0.472857
\(733\) 48.7218 1.79958 0.899790 0.436323i \(-0.143719\pi\)
0.899790 + 0.436323i \(0.143719\pi\)
\(734\) −29.4898 −1.08849
\(735\) 0 0
\(736\) 32.7314 1.20650
\(737\) 7.98016 0.293953
\(738\) −14.2361 −0.524038
\(739\) −9.48405 −0.348876 −0.174438 0.984668i \(-0.555811\pi\)
−0.174438 + 0.984668i \(0.555811\pi\)
\(740\) 0 0
\(741\) −24.2205 −0.889761
\(742\) 0 0
\(743\) 30.2032 1.10805 0.554023 0.832501i \(-0.313092\pi\)
0.554023 + 0.832501i \(0.313092\pi\)
\(744\) −4.28376 −0.157050
\(745\) 0 0
\(746\) −57.2669 −2.09669
\(747\) −8.75128 −0.320192
\(748\) 5.19937 0.190108
\(749\) 0 0
\(750\) 0 0
\(751\) −7.32934 −0.267451 −0.133726 0.991018i \(-0.542694\pi\)
−0.133726 + 0.991018i \(0.542694\pi\)
\(752\) −12.1296 −0.442322
\(753\) −0.311597 −0.0113552
\(754\) −22.3842 −0.815186
\(755\) 0 0
\(756\) 0 0
\(757\) −14.1603 −0.514666 −0.257333 0.966323i \(-0.582844\pi\)
−0.257333 + 0.966323i \(0.582844\pi\)
\(758\) −8.88688 −0.322786
\(759\) −11.5077 −0.417703
\(760\) 0 0
\(761\) 23.9183 0.867039 0.433519 0.901144i \(-0.357272\pi\)
0.433519 + 0.901144i \(0.357272\pi\)
\(762\) 15.8029 0.572477
\(763\) 0 0
\(764\) 13.8665 0.501672
\(765\) 0 0
\(766\) 47.1369 1.70312
\(767\) −9.25278 −0.334099
\(768\) −20.3687 −0.734992
\(769\) −14.1358 −0.509750 −0.254875 0.966974i \(-0.582034\pi\)
−0.254875 + 0.966974i \(0.582034\pi\)
\(770\) 0 0
\(771\) 2.50211 0.0901113
\(772\) −29.0496 −1.04552
\(773\) 6.75972 0.243130 0.121565 0.992583i \(-0.461209\pi\)
0.121565 + 0.992583i \(0.461209\pi\)
\(774\) 8.89113 0.319585
\(775\) 0 0
\(776\) 0.436693 0.0156764
\(777\) 0 0
\(778\) −22.7545 −0.815790
\(779\) −61.8263 −2.21516
\(780\) 0 0
\(781\) 31.2290 1.11746
\(782\) 21.0131 0.751425
\(783\) 4.01368 0.143437
\(784\) 0 0
\(785\) 0 0
\(786\) 9.51671 0.339450
\(787\) 15.3751 0.548062 0.274031 0.961721i \(-0.411643\pi\)
0.274031 + 0.961721i \(0.411643\pi\)
\(788\) −22.9604 −0.817931
\(789\) −4.62511 −0.164658
\(790\) 0 0
\(791\) 0 0
\(792\) 3.05774 0.108652
\(793\) 34.1026 1.21102
\(794\) −26.9408 −0.956092
\(795\) 0 0
\(796\) 26.1164 0.925670
\(797\) −9.46785 −0.335369 −0.167684 0.985841i \(-0.553629\pi\)
−0.167684 + 0.985841i \(0.553629\pi\)
\(798\) 0 0
\(799\) −5.19937 −0.183941
\(800\) 0 0
\(801\) −0.618661 −0.0218593
\(802\) 22.0223 0.777634
\(803\) 17.7788 0.627400
\(804\) 4.50856 0.159004
\(805\) 0 0
\(806\) −16.2395 −0.572014
\(807\) 12.0233 0.423240
\(808\) 12.1498 0.427427
\(809\) 5.63745 0.198202 0.0991011 0.995077i \(-0.468403\pi\)
0.0991011 + 0.995077i \(0.468403\pi\)
\(810\) 0 0
\(811\) −24.3625 −0.855485 −0.427742 0.903901i \(-0.640691\pi\)
−0.427742 + 0.903901i \(0.640691\pi\)
\(812\) 0 0
\(813\) −5.46935 −0.191818
\(814\) 13.0173 0.456256
\(815\) 0 0
\(816\) −10.5864 −0.370600
\(817\) 38.6135 1.35092
\(818\) 18.7585 0.655876
\(819\) 0 0
\(820\) 0 0
\(821\) −31.7462 −1.10795 −0.553974 0.832534i \(-0.686889\pi\)
−0.553974 + 0.832534i \(0.686889\pi\)
\(822\) −23.3466 −0.814307
\(823\) 3.97563 0.138582 0.0692908 0.997597i \(-0.477926\pi\)
0.0692908 + 0.997597i \(0.477926\pi\)
\(824\) 25.1366 0.875677
\(825\) 0 0
\(826\) 0 0
\(827\) −6.80348 −0.236580 −0.118290 0.992979i \(-0.537741\pi\)
−0.118290 + 0.992979i \(0.537741\pi\)
\(828\) −6.50151 −0.225943
\(829\) −23.0606 −0.800927 −0.400463 0.916313i \(-0.631151\pi\)
−0.400463 + 0.916313i \(0.631151\pi\)
\(830\) 0 0
\(831\) 29.8099 1.03410
\(832\) −1.85837 −0.0644273
\(833\) 0 0
\(834\) 0.433235 0.0150017
\(835\) 0 0
\(836\) −18.8856 −0.653171
\(837\) 2.91188 0.100649
\(838\) −23.7155 −0.819239
\(839\) 35.0723 1.21083 0.605415 0.795910i \(-0.293007\pi\)
0.605415 + 0.795910i \(0.293007\pi\)
\(840\) 0 0
\(841\) −12.8904 −0.444495
\(842\) 34.1223 1.17593
\(843\) −7.78511 −0.268133
\(844\) 27.7357 0.954701
\(845\) 0 0
\(846\) 4.34858 0.149507
\(847\) 0 0
\(848\) 49.2583 1.69154
\(849\) 2.61444 0.0897274
\(850\) 0 0
\(851\) 19.4620 0.667149
\(852\) 17.6435 0.604456
\(853\) −12.3125 −0.421571 −0.210785 0.977532i \(-0.567602\pi\)
−0.210785 + 0.977532i \(0.567602\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 9.28360 0.317307
\(857\) 33.2498 1.13579 0.567896 0.823101i \(-0.307758\pi\)
0.567896 + 0.823101i \(0.307758\pi\)
\(858\) 11.5917 0.395736
\(859\) 3.01395 0.102834 0.0514172 0.998677i \(-0.483626\pi\)
0.0514172 + 0.998677i \(0.483626\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 19.4136 0.661228
\(863\) 22.2360 0.756921 0.378460 0.925617i \(-0.376454\pi\)
0.378460 + 0.925617i \(0.376454\pi\)
\(864\) 5.91188 0.201126
\(865\) 0 0
\(866\) 34.7190 1.17980
\(867\) 12.4621 0.423236
\(868\) 0 0
\(869\) −16.8639 −0.572070
\(870\) 0 0
\(871\) −12.0182 −0.407221
\(872\) −3.59950 −0.121894
\(873\) −0.296842 −0.0100466
\(874\) −76.3253 −2.58174
\(875\) 0 0
\(876\) 10.0445 0.339372
\(877\) 14.3221 0.483623 0.241812 0.970323i \(-0.422258\pi\)
0.241812 + 0.970323i \(0.422258\pi\)
\(878\) −24.1075 −0.813590
\(879\) −12.7559 −0.430247
\(880\) 0 0
\(881\) −49.9929 −1.68431 −0.842153 0.539239i \(-0.818712\pi\)
−0.842153 + 0.539239i \(0.818712\pi\)
\(882\) 0 0
\(883\) −44.3095 −1.49113 −0.745566 0.666432i \(-0.767821\pi\)
−0.745566 + 0.666432i \(0.767821\pi\)
\(884\) −7.83030 −0.263361
\(885\) 0 0
\(886\) −51.4298 −1.72782
\(887\) −10.2985 −0.345790 −0.172895 0.984940i \(-0.555312\pi\)
−0.172895 + 0.984940i \(0.555312\pi\)
\(888\) −5.17130 −0.173538
\(889\) 0 0
\(890\) 0 0
\(891\) −2.07850 −0.0696322
\(892\) −29.5239 −0.988535
\(893\) 18.8856 0.631981
\(894\) −4.16423 −0.139273
\(895\) 0 0
\(896\) 0 0
\(897\) 17.3307 0.578654
\(898\) 41.7438 1.39301
\(899\) 11.6874 0.389796
\(900\) 0 0
\(901\) 21.1146 0.703430
\(902\) 29.5896 0.985227
\(903\) 0 0
\(904\) 14.9423 0.496972
\(905\) 0 0
\(906\) 19.8005 0.657827
\(907\) −29.4280 −0.977139 −0.488570 0.872525i \(-0.662481\pi\)
−0.488570 + 0.872525i \(0.662481\pi\)
\(908\) 27.7957 0.922433
\(909\) −8.25879 −0.273927
\(910\) 0 0
\(911\) −24.8078 −0.821918 −0.410959 0.911654i \(-0.634806\pi\)
−0.410959 + 0.911654i \(0.634806\pi\)
\(912\) 38.4529 1.27330
\(913\) 18.1895 0.601984
\(914\) −53.9619 −1.78490
\(915\) 0 0
\(916\) −0.477133 −0.0157649
\(917\) 0 0
\(918\) 3.79533 0.125265
\(919\) 13.0093 0.429138 0.214569 0.976709i \(-0.431165\pi\)
0.214569 + 0.976709i \(0.431165\pi\)
\(920\) 0 0
\(921\) −13.1919 −0.434687
\(922\) 12.5107 0.412018
\(923\) −47.0312 −1.54805
\(924\) 0 0
\(925\) 0 0
\(926\) −5.29729 −0.174080
\(927\) −17.0866 −0.561198
\(928\) 23.7284 0.778924
\(929\) −24.9185 −0.817549 −0.408775 0.912635i \(-0.634044\pi\)
−0.408775 + 0.912635i \(0.634044\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 8.41410 0.275613
\(933\) −14.2823 −0.467580
\(934\) 42.3230 1.38485
\(935\) 0 0
\(936\) −4.60498 −0.150518
\(937\) 55.1260 1.80089 0.900444 0.434973i \(-0.143242\pi\)
0.900444 + 0.434973i \(0.143242\pi\)
\(938\) 0 0
\(939\) 4.77143 0.155710
\(940\) 0 0
\(941\) −13.6447 −0.444803 −0.222402 0.974955i \(-0.571390\pi\)
−0.222402 + 0.974955i \(0.571390\pi\)
\(942\) −20.1651 −0.657015
\(943\) 44.2391 1.44062
\(944\) 14.6899 0.478116
\(945\) 0 0
\(946\) −18.4802 −0.600842
\(947\) −0.893089 −0.0290215 −0.0145107 0.999895i \(-0.504619\pi\)
−0.0145107 + 0.999895i \(0.504619\pi\)
\(948\) −9.52764 −0.309443
\(949\) −26.7750 −0.869154
\(950\) 0 0
\(951\) 18.8048 0.609787
\(952\) 0 0
\(953\) 34.5636 1.11963 0.559813 0.828619i \(-0.310873\pi\)
0.559813 + 0.828619i \(0.310873\pi\)
\(954\) −17.6595 −0.571749
\(955\) 0 0
\(956\) 11.8511 0.383293
\(957\) −8.34242 −0.269672
\(958\) 11.6595 0.376703
\(959\) 0 0
\(960\) 0 0
\(961\) −22.5209 −0.726481
\(962\) −19.6042 −0.632064
\(963\) −6.31052 −0.203354
\(964\) 5.40967 0.174234
\(965\) 0 0
\(966\) 0 0
\(967\) 22.1811 0.713296 0.356648 0.934239i \(-0.383920\pi\)
0.356648 + 0.934239i \(0.383920\pi\)
\(968\) 9.82694 0.315850
\(969\) 16.4829 0.529506
\(970\) 0 0
\(971\) −0.0759319 −0.00243677 −0.00121839 0.999999i \(-0.500388\pi\)
−0.00121839 + 0.999999i \(0.500388\pi\)
\(972\) −1.17429 −0.0376653
\(973\) 0 0
\(974\) −29.0889 −0.932069
\(975\) 0 0
\(976\) −54.1420 −1.73304
\(977\) −43.2114 −1.38246 −0.691228 0.722636i \(-0.742930\pi\)
−0.691228 + 0.722636i \(0.742930\pi\)
\(978\) −3.74766 −0.119837
\(979\) 1.28588 0.0410970
\(980\) 0 0
\(981\) 2.44676 0.0781189
\(982\) 44.3648 1.41574
\(983\) 18.3208 0.584343 0.292171 0.956366i \(-0.405622\pi\)
0.292171 + 0.956366i \(0.405622\pi\)
\(984\) −11.7549 −0.374732
\(985\) 0 0
\(986\) 15.2333 0.485126
\(987\) 0 0
\(988\) 28.4418 0.904855
\(989\) −27.6295 −0.878566
\(990\) 0 0
\(991\) −31.5776 −1.00310 −0.501548 0.865130i \(-0.667236\pi\)
−0.501548 + 0.865130i \(0.667236\pi\)
\(992\) 17.2147 0.546568
\(993\) −17.7742 −0.564046
\(994\) 0 0
\(995\) 0 0
\(996\) 10.2765 0.325624
\(997\) −39.7315 −1.25831 −0.629154 0.777281i \(-0.716599\pi\)
−0.629154 + 0.777281i \(0.716599\pi\)
\(998\) −21.7432 −0.688268
\(999\) 3.51519 0.111216
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 3675.2.a.bn.1.4 4
5.2 odd 4 735.2.d.e.589.7 8
5.3 odd 4 735.2.d.e.589.2 8
5.4 even 2 3675.2.a.cb.1.1 4
7.3 odd 6 525.2.i.k.226.1 8
7.5 odd 6 525.2.i.k.151.1 8
7.6 odd 2 3675.2.a.bp.1.4 4
15.2 even 4 2205.2.d.o.1324.2 8
15.8 even 4 2205.2.d.o.1324.7 8
35.2 odd 12 735.2.q.g.214.7 16
35.3 even 12 105.2.q.a.79.7 yes 16
35.12 even 12 105.2.q.a.4.7 yes 16
35.13 even 4 735.2.d.d.589.2 8
35.17 even 12 105.2.q.a.79.2 yes 16
35.18 odd 12 735.2.q.g.79.7 16
35.19 odd 6 525.2.i.h.151.4 8
35.23 odd 12 735.2.q.g.214.2 16
35.24 odd 6 525.2.i.h.226.4 8
35.27 even 4 735.2.d.d.589.7 8
35.32 odd 12 735.2.q.g.79.2 16
35.33 even 12 105.2.q.a.4.2 16
35.34 odd 2 3675.2.a.bz.1.1 4
105.17 odd 12 315.2.bf.b.289.7 16
105.38 odd 12 315.2.bf.b.289.2 16
105.47 odd 12 315.2.bf.b.109.2 16
105.62 odd 4 2205.2.d.s.1324.2 8
105.68 odd 12 315.2.bf.b.109.7 16
105.83 odd 4 2205.2.d.s.1324.7 8
140.3 odd 12 1680.2.di.d.289.2 16
140.47 odd 12 1680.2.di.d.529.2 16
140.87 odd 12 1680.2.di.d.289.6 16
140.103 odd 12 1680.2.di.d.529.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.q.a.4.2 16 35.33 even 12
105.2.q.a.4.7 yes 16 35.12 even 12
105.2.q.a.79.2 yes 16 35.17 even 12
105.2.q.a.79.7 yes 16 35.3 even 12
315.2.bf.b.109.2 16 105.47 odd 12
315.2.bf.b.109.7 16 105.68 odd 12
315.2.bf.b.289.2 16 105.38 odd 12
315.2.bf.b.289.7 16 105.17 odd 12
525.2.i.h.151.4 8 35.19 odd 6
525.2.i.h.226.4 8 35.24 odd 6
525.2.i.k.151.1 8 7.5 odd 6
525.2.i.k.226.1 8 7.3 odd 6
735.2.d.d.589.2 8 35.13 even 4
735.2.d.d.589.7 8 35.27 even 4
735.2.d.e.589.2 8 5.3 odd 4
735.2.d.e.589.7 8 5.2 odd 4
735.2.q.g.79.2 16 35.32 odd 12
735.2.q.g.79.7 16 35.18 odd 12
735.2.q.g.214.2 16 35.23 odd 12
735.2.q.g.214.7 16 35.2 odd 12
1680.2.di.d.289.2 16 140.3 odd 12
1680.2.di.d.289.6 16 140.87 odd 12
1680.2.di.d.529.2 16 140.47 odd 12
1680.2.di.d.529.6 16 140.103 odd 12
2205.2.d.o.1324.2 8 15.2 even 4
2205.2.d.o.1324.7 8 15.8 even 4
2205.2.d.s.1324.2 8 105.62 odd 4
2205.2.d.s.1324.7 8 105.83 odd 4
3675.2.a.bn.1.4 4 1.1 even 1 trivial
3675.2.a.bp.1.4 4 7.6 odd 2
3675.2.a.bz.1.1 4 35.34 odd 2
3675.2.a.cb.1.1 4 5.4 even 2