Properties

Label 5225.2.a.y.1.10
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $0$
Dimension $15$
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] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, 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("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 5 x^{14} - 11 x^{13} + 87 x^{12} - 4 x^{11} - 545 x^{10} + 431 x^{9} + 1480 x^{8} - 1763 x^{7} + \cdots + 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(1.28989\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.28989 q^{2} +3.30063 q^{3} -0.336174 q^{4} +4.25747 q^{6} +0.563541 q^{7} -3.01342 q^{8} +7.89418 q^{9} +O(q^{10})\) \(q+1.28989 q^{2} +3.30063 q^{3} -0.336174 q^{4} +4.25747 q^{6} +0.563541 q^{7} -3.01342 q^{8} +7.89418 q^{9} +1.00000 q^{11} -1.10959 q^{12} -1.69967 q^{13} +0.726907 q^{14} -3.21464 q^{16} -0.0949420 q^{17} +10.1826 q^{18} +1.00000 q^{19} +1.86004 q^{21} +1.28989 q^{22} +0.625186 q^{23} -9.94618 q^{24} -2.19239 q^{26} +16.1539 q^{27} -0.189448 q^{28} +6.14313 q^{29} +3.11276 q^{31} +1.88029 q^{32} +3.30063 q^{33} -0.122465 q^{34} -2.65382 q^{36} +2.91877 q^{37} +1.28989 q^{38} -5.60998 q^{39} +0.562373 q^{41} +2.39925 q^{42} +10.9758 q^{43} -0.336174 q^{44} +0.806423 q^{46} +8.90408 q^{47} -10.6103 q^{48} -6.68242 q^{49} -0.313369 q^{51} +0.571385 q^{52} -0.489328 q^{53} +20.8368 q^{54} -1.69818 q^{56} +3.30063 q^{57} +7.92399 q^{58} -11.5958 q^{59} +13.8576 q^{61} +4.01513 q^{62} +4.44869 q^{63} +8.85465 q^{64} +4.25747 q^{66} -2.05662 q^{67} +0.0319171 q^{68} +2.06351 q^{69} +3.06337 q^{71} -23.7884 q^{72} -5.61322 q^{73} +3.76490 q^{74} -0.336174 q^{76} +0.563541 q^{77} -7.23628 q^{78} -5.37103 q^{79} +29.6355 q^{81} +0.725401 q^{82} +2.47369 q^{83} -0.625298 q^{84} +14.1576 q^{86} +20.2762 q^{87} -3.01342 q^{88} -14.0050 q^{89} -0.957832 q^{91} -0.210171 q^{92} +10.2741 q^{93} +11.4853 q^{94} +6.20615 q^{96} +6.13801 q^{97} -8.61961 q^{98} +7.89418 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 21 q^{7} + 9 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 21 q^{7} + 9 q^{8} + 15 q^{9} + 15 q^{11} + 11 q^{12} + 13 q^{13} + 9 q^{14} + 21 q^{16} + 17 q^{17} + 22 q^{18} + 15 q^{19} + 6 q^{21} + 5 q^{22} + 26 q^{23} + q^{24} + 3 q^{26} + q^{27} + 46 q^{28} + 9 q^{29} + 14 q^{31} + 18 q^{32} + 4 q^{33} - 13 q^{34} + 12 q^{36} + 9 q^{37} + 5 q^{38} - 22 q^{39} + 4 q^{41} - 6 q^{42} + 28 q^{43} + 17 q^{44} + 27 q^{46} + 14 q^{47} - 4 q^{48} + 32 q^{49} - 40 q^{51} + 14 q^{52} + 3 q^{53} - 39 q^{54} + 34 q^{56} + 4 q^{57} + 26 q^{58} + q^{59} + 2 q^{61} - 3 q^{62} + 45 q^{63} + 5 q^{64} - q^{66} + 37 q^{67} + 26 q^{68} - 7 q^{69} - 7 q^{71} + 16 q^{72} + 42 q^{73} - 43 q^{74} + 17 q^{76} + 21 q^{77} - 64 q^{78} - 10 q^{79} + 31 q^{81} + 22 q^{82} + 14 q^{83} - 32 q^{84} + 37 q^{86} + 29 q^{87} + 9 q^{88} + 15 q^{89} - 22 q^{91} + 26 q^{92} - 18 q^{93} - 44 q^{94} + 71 q^{96} + 8 q^{97} - 10 q^{98} + 15 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.28989 0.912093 0.456046 0.889956i \(-0.349265\pi\)
0.456046 + 0.889956i \(0.349265\pi\)
\(3\) 3.30063 1.90562 0.952811 0.303565i \(-0.0981771\pi\)
0.952811 + 0.303565i \(0.0981771\pi\)
\(4\) −0.336174 −0.168087
\(5\) 0 0
\(6\) 4.25747 1.73810
\(7\) 0.563541 0.212998 0.106499 0.994313i \(-0.466036\pi\)
0.106499 + 0.994313i \(0.466036\pi\)
\(8\) −3.01342 −1.06540
\(9\) 7.89418 2.63139
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −1.10959 −0.320310
\(13\) −1.69967 −0.471403 −0.235702 0.971825i \(-0.575739\pi\)
−0.235702 + 0.971825i \(0.575739\pi\)
\(14\) 0.726907 0.194274
\(15\) 0 0
\(16\) −3.21464 −0.803660
\(17\) −0.0949420 −0.0230268 −0.0115134 0.999934i \(-0.503665\pi\)
−0.0115134 + 0.999934i \(0.503665\pi\)
\(18\) 10.1826 2.40007
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 1.86004 0.405894
\(22\) 1.28989 0.275006
\(23\) 0.625186 0.130360 0.0651801 0.997874i \(-0.479238\pi\)
0.0651801 + 0.997874i \(0.479238\pi\)
\(24\) −9.94618 −2.03026
\(25\) 0 0
\(26\) −2.19239 −0.429963
\(27\) 16.1539 3.10882
\(28\) −0.189448 −0.0358023
\(29\) 6.14313 1.14075 0.570376 0.821384i \(-0.306798\pi\)
0.570376 + 0.821384i \(0.306798\pi\)
\(30\) 0 0
\(31\) 3.11276 0.559069 0.279534 0.960136i \(-0.409820\pi\)
0.279534 + 0.960136i \(0.409820\pi\)
\(32\) 1.88029 0.332392
\(33\) 3.30063 0.574566
\(34\) −0.122465 −0.0210026
\(35\) 0 0
\(36\) −2.65382 −0.442303
\(37\) 2.91877 0.479842 0.239921 0.970792i \(-0.422878\pi\)
0.239921 + 0.970792i \(0.422878\pi\)
\(38\) 1.28989 0.209248
\(39\) −5.60998 −0.898316
\(40\) 0 0
\(41\) 0.562373 0.0878279 0.0439139 0.999035i \(-0.486017\pi\)
0.0439139 + 0.999035i \(0.486017\pi\)
\(42\) 2.39925 0.370213
\(43\) 10.9758 1.67379 0.836895 0.547364i \(-0.184369\pi\)
0.836895 + 0.547364i \(0.184369\pi\)
\(44\) −0.336174 −0.0506802
\(45\) 0 0
\(46\) 0.806423 0.118901
\(47\) 8.90408 1.29879 0.649396 0.760450i \(-0.275022\pi\)
0.649396 + 0.760450i \(0.275022\pi\)
\(48\) −10.6103 −1.53147
\(49\) −6.68242 −0.954632
\(50\) 0 0
\(51\) −0.313369 −0.0438804
\(52\) 0.571385 0.0792368
\(53\) −0.489328 −0.0672144 −0.0336072 0.999435i \(-0.510700\pi\)
−0.0336072 + 0.999435i \(0.510700\pi\)
\(54\) 20.8368 2.83553
\(55\) 0 0
\(56\) −1.69818 −0.226929
\(57\) 3.30063 0.437179
\(58\) 7.92399 1.04047
\(59\) −11.5958 −1.50965 −0.754824 0.655927i \(-0.772278\pi\)
−0.754824 + 0.655927i \(0.772278\pi\)
\(60\) 0 0
\(61\) 13.8576 1.77428 0.887139 0.461502i \(-0.152689\pi\)
0.887139 + 0.461502i \(0.152689\pi\)
\(62\) 4.01513 0.509922
\(63\) 4.44869 0.560482
\(64\) 8.85465 1.10683
\(65\) 0 0
\(66\) 4.25747 0.524058
\(67\) −2.05662 −0.251257 −0.125628 0.992077i \(-0.540095\pi\)
−0.125628 + 0.992077i \(0.540095\pi\)
\(68\) 0.0319171 0.00387051
\(69\) 2.06351 0.248417
\(70\) 0 0
\(71\) 3.06337 0.363556 0.181778 0.983340i \(-0.441815\pi\)
0.181778 + 0.983340i \(0.441815\pi\)
\(72\) −23.7884 −2.80349
\(73\) −5.61322 −0.656978 −0.328489 0.944508i \(-0.606539\pi\)
−0.328489 + 0.944508i \(0.606539\pi\)
\(74\) 3.76490 0.437661
\(75\) 0 0
\(76\) −0.336174 −0.0385618
\(77\) 0.563541 0.0642214
\(78\) −7.23628 −0.819347
\(79\) −5.37103 −0.604288 −0.302144 0.953262i \(-0.597702\pi\)
−0.302144 + 0.953262i \(0.597702\pi\)
\(80\) 0 0
\(81\) 29.6355 3.29283
\(82\) 0.725401 0.0801071
\(83\) 2.47369 0.271522 0.135761 0.990742i \(-0.456652\pi\)
0.135761 + 0.990742i \(0.456652\pi\)
\(84\) −0.625298 −0.0682256
\(85\) 0 0
\(86\) 14.1576 1.52665
\(87\) 20.2762 2.17384
\(88\) −3.01342 −0.321231
\(89\) −14.0050 −1.48452 −0.742262 0.670110i \(-0.766247\pi\)
−0.742262 + 0.670110i \(0.766247\pi\)
\(90\) 0 0
\(91\) −0.957832 −0.100408
\(92\) −0.210171 −0.0219119
\(93\) 10.2741 1.06537
\(94\) 11.4853 1.18462
\(95\) 0 0
\(96\) 6.20615 0.633413
\(97\) 6.13801 0.623221 0.311610 0.950210i \(-0.399132\pi\)
0.311610 + 0.950210i \(0.399132\pi\)
\(98\) −8.61961 −0.870712
\(99\) 7.89418 0.793395
\(100\) 0 0
\(101\) 12.4177 1.23561 0.617804 0.786332i \(-0.288023\pi\)
0.617804 + 0.786332i \(0.288023\pi\)
\(102\) −0.404212 −0.0400230
\(103\) −10.7035 −1.05465 −0.527324 0.849665i \(-0.676804\pi\)
−0.527324 + 0.849665i \(0.676804\pi\)
\(104\) 5.12181 0.502234
\(105\) 0 0
\(106\) −0.631181 −0.0613057
\(107\) 2.53215 0.244792 0.122396 0.992481i \(-0.460942\pi\)
0.122396 + 0.992481i \(0.460942\pi\)
\(108\) −5.43052 −0.522552
\(109\) −10.2842 −0.985046 −0.492523 0.870299i \(-0.663925\pi\)
−0.492523 + 0.870299i \(0.663925\pi\)
\(110\) 0 0
\(111\) 9.63378 0.914398
\(112\) −1.81158 −0.171178
\(113\) −17.0132 −1.60047 −0.800236 0.599686i \(-0.795292\pi\)
−0.800236 + 0.599686i \(0.795292\pi\)
\(114\) 4.25747 0.398748
\(115\) 0 0
\(116\) −2.06516 −0.191746
\(117\) −13.4175 −1.24045
\(118\) −14.9574 −1.37694
\(119\) −0.0535037 −0.00490468
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 17.8748 1.61831
\(123\) 1.85619 0.167367
\(124\) −1.04643 −0.0939723
\(125\) 0 0
\(126\) 5.73834 0.511212
\(127\) −5.76757 −0.511789 −0.255894 0.966705i \(-0.582370\pi\)
−0.255894 + 0.966705i \(0.582370\pi\)
\(128\) 7.66098 0.677141
\(129\) 36.2270 3.18961
\(130\) 0 0
\(131\) −6.29905 −0.550351 −0.275175 0.961394i \(-0.588736\pi\)
−0.275175 + 0.961394i \(0.588736\pi\)
\(132\) −1.10959 −0.0965772
\(133\) 0.563541 0.0488652
\(134\) −2.65283 −0.229169
\(135\) 0 0
\(136\) 0.286100 0.0245329
\(137\) 9.89969 0.845787 0.422894 0.906179i \(-0.361014\pi\)
0.422894 + 0.906179i \(0.361014\pi\)
\(138\) 2.66171 0.226579
\(139\) 21.4383 1.81838 0.909188 0.416387i \(-0.136704\pi\)
0.909188 + 0.416387i \(0.136704\pi\)
\(140\) 0 0
\(141\) 29.3891 2.47501
\(142\) 3.95143 0.331596
\(143\) −1.69967 −0.142133
\(144\) −25.3769 −2.11474
\(145\) 0 0
\(146\) −7.24046 −0.599225
\(147\) −22.0562 −1.81917
\(148\) −0.981214 −0.0806553
\(149\) −5.05856 −0.414413 −0.207207 0.978297i \(-0.566437\pi\)
−0.207207 + 0.978297i \(0.566437\pi\)
\(150\) 0 0
\(151\) −13.4546 −1.09492 −0.547460 0.836832i \(-0.684405\pi\)
−0.547460 + 0.836832i \(0.684405\pi\)
\(152\) −3.01342 −0.244420
\(153\) −0.749489 −0.0605926
\(154\) 0.726907 0.0585759
\(155\) 0 0
\(156\) 1.88593 0.150995
\(157\) 2.56715 0.204881 0.102440 0.994739i \(-0.467335\pi\)
0.102440 + 0.994739i \(0.467335\pi\)
\(158\) −6.92806 −0.551167
\(159\) −1.61509 −0.128085
\(160\) 0 0
\(161\) 0.352317 0.0277665
\(162\) 38.2266 3.00337
\(163\) −14.3323 −1.12259 −0.561294 0.827616i \(-0.689696\pi\)
−0.561294 + 0.827616i \(0.689696\pi\)
\(164\) −0.189055 −0.0147627
\(165\) 0 0
\(166\) 3.19079 0.247654
\(167\) −16.9732 −1.31342 −0.656711 0.754142i \(-0.728053\pi\)
−0.656711 + 0.754142i \(0.728053\pi\)
\(168\) −5.60508 −0.432441
\(169\) −10.1111 −0.777779
\(170\) 0 0
\(171\) 7.89418 0.603683
\(172\) −3.68977 −0.281343
\(173\) −24.9706 −1.89848 −0.949241 0.314549i \(-0.898147\pi\)
−0.949241 + 0.314549i \(0.898147\pi\)
\(174\) 26.1542 1.98274
\(175\) 0 0
\(176\) −3.21464 −0.242312
\(177\) −38.2736 −2.87682
\(178\) −18.0649 −1.35402
\(179\) 2.25582 0.168608 0.0843040 0.996440i \(-0.473133\pi\)
0.0843040 + 0.996440i \(0.473133\pi\)
\(180\) 0 0
\(181\) 5.14529 0.382446 0.191223 0.981547i \(-0.438755\pi\)
0.191223 + 0.981547i \(0.438755\pi\)
\(182\) −1.23550 −0.0915814
\(183\) 45.7387 3.38110
\(184\) −1.88394 −0.138886
\(185\) 0 0
\(186\) 13.2525 0.971719
\(187\) −0.0949420 −0.00694285
\(188\) −2.99332 −0.218310
\(189\) 9.10337 0.662172
\(190\) 0 0
\(191\) −1.23491 −0.0893549 −0.0446774 0.999001i \(-0.514226\pi\)
−0.0446774 + 0.999001i \(0.514226\pi\)
\(192\) 29.2260 2.10920
\(193\) −6.39165 −0.460081 −0.230041 0.973181i \(-0.573886\pi\)
−0.230041 + 0.973181i \(0.573886\pi\)
\(194\) 7.91738 0.568435
\(195\) 0 0
\(196\) 2.24646 0.160461
\(197\) 5.23212 0.372773 0.186387 0.982476i \(-0.440322\pi\)
0.186387 + 0.982476i \(0.440322\pi\)
\(198\) 10.1826 0.723649
\(199\) −19.1059 −1.35438 −0.677192 0.735807i \(-0.736803\pi\)
−0.677192 + 0.735807i \(0.736803\pi\)
\(200\) 0 0
\(201\) −6.78816 −0.478800
\(202\) 16.0175 1.12699
\(203\) 3.46190 0.242978
\(204\) 0.105347 0.00737573
\(205\) 0 0
\(206\) −13.8064 −0.961936
\(207\) 4.93532 0.343029
\(208\) 5.46382 0.378848
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) −3.42015 −0.235453 −0.117726 0.993046i \(-0.537561\pi\)
−0.117726 + 0.993046i \(0.537561\pi\)
\(212\) 0.164499 0.0112979
\(213\) 10.1111 0.692799
\(214\) 3.26621 0.223273
\(215\) 0 0
\(216\) −48.6784 −3.31214
\(217\) 1.75417 0.119081
\(218\) −13.2655 −0.898453
\(219\) −18.5272 −1.25195
\(220\) 0 0
\(221\) 0.161370 0.0108549
\(222\) 12.4265 0.834015
\(223\) −10.6147 −0.710813 −0.355406 0.934712i \(-0.615657\pi\)
−0.355406 + 0.934712i \(0.615657\pi\)
\(224\) 1.05962 0.0707989
\(225\) 0 0
\(226\) −21.9453 −1.45978
\(227\) 7.18967 0.477195 0.238598 0.971119i \(-0.423312\pi\)
0.238598 + 0.971119i \(0.423312\pi\)
\(228\) −1.10959 −0.0734843
\(229\) −21.1961 −1.40067 −0.700337 0.713812i \(-0.746967\pi\)
−0.700337 + 0.713812i \(0.746967\pi\)
\(230\) 0 0
\(231\) 1.86004 0.122382
\(232\) −18.5118 −1.21536
\(233\) 26.7869 1.75487 0.877433 0.479699i \(-0.159254\pi\)
0.877433 + 0.479699i \(0.159254\pi\)
\(234\) −17.3071 −1.13140
\(235\) 0 0
\(236\) 3.89822 0.253752
\(237\) −17.7278 −1.15154
\(238\) −0.0690141 −0.00447352
\(239\) 5.29366 0.342419 0.171209 0.985235i \(-0.445233\pi\)
0.171209 + 0.985235i \(0.445233\pi\)
\(240\) 0 0
\(241\) 1.54532 0.0995429 0.0497714 0.998761i \(-0.484151\pi\)
0.0497714 + 0.998761i \(0.484151\pi\)
\(242\) 1.28989 0.0829175
\(243\) 49.3542 3.16608
\(244\) −4.65856 −0.298233
\(245\) 0 0
\(246\) 2.39428 0.152654
\(247\) −1.69967 −0.108147
\(248\) −9.38005 −0.595634
\(249\) 8.16473 0.517419
\(250\) 0 0
\(251\) −23.7169 −1.49700 −0.748500 0.663135i \(-0.769226\pi\)
−0.748500 + 0.663135i \(0.769226\pi\)
\(252\) −1.49553 −0.0942098
\(253\) 0.625186 0.0393051
\(254\) −7.43955 −0.466799
\(255\) 0 0
\(256\) −7.82746 −0.489216
\(257\) −2.55273 −0.159235 −0.0796174 0.996825i \(-0.525370\pi\)
−0.0796174 + 0.996825i \(0.525370\pi\)
\(258\) 46.7290 2.90922
\(259\) 1.64484 0.102206
\(260\) 0 0
\(261\) 48.4950 3.00176
\(262\) −8.12511 −0.501971
\(263\) −4.06910 −0.250911 −0.125456 0.992099i \(-0.540039\pi\)
−0.125456 + 0.992099i \(0.540039\pi\)
\(264\) −9.94618 −0.612145
\(265\) 0 0
\(266\) 0.726907 0.0445696
\(267\) −46.2252 −2.82894
\(268\) 0.691384 0.0422330
\(269\) 0.181743 0.0110811 0.00554055 0.999985i \(-0.498236\pi\)
0.00554055 + 0.999985i \(0.498236\pi\)
\(270\) 0 0
\(271\) −4.15652 −0.252491 −0.126245 0.991999i \(-0.540293\pi\)
−0.126245 + 0.991999i \(0.540293\pi\)
\(272\) 0.305204 0.0185057
\(273\) −3.16145 −0.191340
\(274\) 12.7695 0.771436
\(275\) 0 0
\(276\) −0.693698 −0.0417557
\(277\) 31.6346 1.90074 0.950368 0.311127i \(-0.100706\pi\)
0.950368 + 0.311127i \(0.100706\pi\)
\(278\) 27.6532 1.65853
\(279\) 24.5727 1.47113
\(280\) 0 0
\(281\) 1.38352 0.0825337 0.0412668 0.999148i \(-0.486861\pi\)
0.0412668 + 0.999148i \(0.486861\pi\)
\(282\) 37.9088 2.25744
\(283\) 25.1918 1.49750 0.748749 0.662854i \(-0.230655\pi\)
0.748749 + 0.662854i \(0.230655\pi\)
\(284\) −1.02983 −0.0611090
\(285\) 0 0
\(286\) −2.19239 −0.129639
\(287\) 0.316920 0.0187072
\(288\) 14.8434 0.874653
\(289\) −16.9910 −0.999470
\(290\) 0 0
\(291\) 20.2593 1.18762
\(292\) 1.88702 0.110430
\(293\) 11.8433 0.691895 0.345947 0.938254i \(-0.387558\pi\)
0.345947 + 0.938254i \(0.387558\pi\)
\(294\) −28.4502 −1.65925
\(295\) 0 0
\(296\) −8.79546 −0.511226
\(297\) 16.1539 0.937343
\(298\) −6.52500 −0.377983
\(299\) −1.06261 −0.0614522
\(300\) 0 0
\(301\) 6.18529 0.356514
\(302\) −17.3550 −0.998668
\(303\) 40.9863 2.35460
\(304\) −3.21464 −0.184372
\(305\) 0 0
\(306\) −0.966761 −0.0552661
\(307\) −12.4995 −0.713385 −0.356692 0.934222i \(-0.616096\pi\)
−0.356692 + 0.934222i \(0.616096\pi\)
\(308\) −0.189448 −0.0107948
\(309\) −35.3283 −2.00976
\(310\) 0 0
\(311\) −4.01056 −0.227418 −0.113709 0.993514i \(-0.536273\pi\)
−0.113709 + 0.993514i \(0.536273\pi\)
\(312\) 16.9052 0.957069
\(313\) 19.8467 1.12180 0.560901 0.827883i \(-0.310455\pi\)
0.560901 + 0.827883i \(0.310455\pi\)
\(314\) 3.31135 0.186870
\(315\) 0 0
\(316\) 1.80560 0.101573
\(317\) −16.2627 −0.913406 −0.456703 0.889619i \(-0.650970\pi\)
−0.456703 + 0.889619i \(0.650970\pi\)
\(318\) −2.08330 −0.116825
\(319\) 6.14313 0.343949
\(320\) 0 0
\(321\) 8.35770 0.466481
\(322\) 0.454452 0.0253256
\(323\) −0.0949420 −0.00528272
\(324\) −9.96269 −0.553483
\(325\) 0 0
\(326\) −18.4871 −1.02390
\(327\) −33.9443 −1.87712
\(328\) −1.69466 −0.0935721
\(329\) 5.01781 0.276641
\(330\) 0 0
\(331\) 17.4946 0.961593 0.480796 0.876832i \(-0.340348\pi\)
0.480796 + 0.876832i \(0.340348\pi\)
\(332\) −0.831590 −0.0456394
\(333\) 23.0413 1.26265
\(334\) −21.8936 −1.19796
\(335\) 0 0
\(336\) −5.97936 −0.326201
\(337\) −34.0435 −1.85447 −0.927234 0.374482i \(-0.877821\pi\)
−0.927234 + 0.374482i \(0.877821\pi\)
\(338\) −13.0423 −0.709407
\(339\) −56.1545 −3.04989
\(340\) 0 0
\(341\) 3.11276 0.168566
\(342\) 10.1826 0.550615
\(343\) −7.71060 −0.416333
\(344\) −33.0746 −1.78326
\(345\) 0 0
\(346\) −32.2095 −1.73159
\(347\) 18.8146 1.01002 0.505011 0.863113i \(-0.331488\pi\)
0.505011 + 0.863113i \(0.331488\pi\)
\(348\) −6.81635 −0.365394
\(349\) 26.8633 1.43796 0.718981 0.695030i \(-0.244609\pi\)
0.718981 + 0.695030i \(0.244609\pi\)
\(350\) 0 0
\(351\) −27.4562 −1.46551
\(352\) 1.88029 0.100220
\(353\) −8.92277 −0.474911 −0.237456 0.971398i \(-0.576313\pi\)
−0.237456 + 0.971398i \(0.576313\pi\)
\(354\) −49.3688 −2.62392
\(355\) 0 0
\(356\) 4.70811 0.249529
\(357\) −0.176596 −0.00934645
\(358\) 2.90977 0.153786
\(359\) 4.83871 0.255377 0.127689 0.991814i \(-0.459244\pi\)
0.127689 + 0.991814i \(0.459244\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 6.63688 0.348826
\(363\) 3.30063 0.173238
\(364\) 0.321998 0.0168773
\(365\) 0 0
\(366\) 58.9981 3.08388
\(367\) 17.7897 0.928614 0.464307 0.885674i \(-0.346303\pi\)
0.464307 + 0.885674i \(0.346303\pi\)
\(368\) −2.00975 −0.104765
\(369\) 4.43947 0.231110
\(370\) 0 0
\(371\) −0.275756 −0.0143165
\(372\) −3.45388 −0.179076
\(373\) 6.12381 0.317079 0.158539 0.987353i \(-0.449322\pi\)
0.158539 + 0.987353i \(0.449322\pi\)
\(374\) −0.122465 −0.00633252
\(375\) 0 0
\(376\) −26.8317 −1.38374
\(377\) −10.4413 −0.537753
\(378\) 11.7424 0.603963
\(379\) −24.5784 −1.26251 −0.631255 0.775575i \(-0.717460\pi\)
−0.631255 + 0.775575i \(0.717460\pi\)
\(380\) 0 0
\(381\) −19.0366 −0.975276
\(382\) −1.59290 −0.0814999
\(383\) −29.3567 −1.50006 −0.750029 0.661405i \(-0.769960\pi\)
−0.750029 + 0.661405i \(0.769960\pi\)
\(384\) 25.2861 1.29037
\(385\) 0 0
\(386\) −8.24455 −0.419637
\(387\) 86.6447 4.40440
\(388\) −2.06344 −0.104755
\(389\) 23.3372 1.18324 0.591621 0.806216i \(-0.298488\pi\)
0.591621 + 0.806216i \(0.298488\pi\)
\(390\) 0 0
\(391\) −0.0593564 −0.00300178
\(392\) 20.1369 1.01707
\(393\) −20.7909 −1.04876
\(394\) 6.74888 0.340004
\(395\) 0 0
\(396\) −2.65382 −0.133359
\(397\) −14.4695 −0.726202 −0.363101 0.931750i \(-0.618282\pi\)
−0.363101 + 0.931750i \(0.618282\pi\)
\(398\) −24.6446 −1.23532
\(399\) 1.86004 0.0931185
\(400\) 0 0
\(401\) −12.7693 −0.637668 −0.318834 0.947811i \(-0.603291\pi\)
−0.318834 + 0.947811i \(0.603291\pi\)
\(402\) −8.75600 −0.436710
\(403\) −5.29066 −0.263547
\(404\) −4.17451 −0.207690
\(405\) 0 0
\(406\) 4.46549 0.221618
\(407\) 2.91877 0.144678
\(408\) 0.944311 0.0467504
\(409\) −23.0952 −1.14198 −0.570991 0.820956i \(-0.693441\pi\)
−0.570991 + 0.820956i \(0.693441\pi\)
\(410\) 0 0
\(411\) 32.6752 1.61175
\(412\) 3.59824 0.177273
\(413\) −6.53472 −0.321552
\(414\) 6.36604 0.312874
\(415\) 0 0
\(416\) −3.19587 −0.156690
\(417\) 70.7601 3.46513
\(418\) 1.28989 0.0630908
\(419\) −2.04650 −0.0999783 −0.0499891 0.998750i \(-0.515919\pi\)
−0.0499891 + 0.998750i \(0.515919\pi\)
\(420\) 0 0
\(421\) 4.36047 0.212516 0.106258 0.994339i \(-0.466113\pi\)
0.106258 + 0.994339i \(0.466113\pi\)
\(422\) −4.41163 −0.214755
\(423\) 70.2903 3.41763
\(424\) 1.47455 0.0716104
\(425\) 0 0
\(426\) 13.0422 0.631897
\(427\) 7.80930 0.377918
\(428\) −0.851244 −0.0411464
\(429\) −5.60998 −0.270852
\(430\) 0 0
\(431\) 15.2001 0.732165 0.366083 0.930582i \(-0.380699\pi\)
0.366083 + 0.930582i \(0.380699\pi\)
\(432\) −51.9289 −2.49843
\(433\) 14.5469 0.699081 0.349541 0.936921i \(-0.386338\pi\)
0.349541 + 0.936921i \(0.386338\pi\)
\(434\) 2.26269 0.108613
\(435\) 0 0
\(436\) 3.45728 0.165574
\(437\) 0.625186 0.0299067
\(438\) −23.8981 −1.14189
\(439\) 34.3065 1.63736 0.818680 0.574250i \(-0.194706\pi\)
0.818680 + 0.574250i \(0.194706\pi\)
\(440\) 0 0
\(441\) −52.7522 −2.51201
\(442\) 0.208150 0.00990069
\(443\) 1.56134 0.0741813 0.0370907 0.999312i \(-0.488191\pi\)
0.0370907 + 0.999312i \(0.488191\pi\)
\(444\) −3.23863 −0.153699
\(445\) 0 0
\(446\) −13.6918 −0.648327
\(447\) −16.6964 −0.789715
\(448\) 4.98996 0.235753
\(449\) −20.7918 −0.981228 −0.490614 0.871377i \(-0.663227\pi\)
−0.490614 + 0.871377i \(0.663227\pi\)
\(450\) 0 0
\(451\) 0.562373 0.0264811
\(452\) 5.71942 0.269019
\(453\) −44.4087 −2.08650
\(454\) 9.27391 0.435246
\(455\) 0 0
\(456\) −9.94618 −0.465773
\(457\) 3.00835 0.140725 0.0703624 0.997521i \(-0.477584\pi\)
0.0703624 + 0.997521i \(0.477584\pi\)
\(458\) −27.3407 −1.27755
\(459\) −1.53368 −0.0715862
\(460\) 0 0
\(461\) 16.5782 0.772122 0.386061 0.922473i \(-0.373835\pi\)
0.386061 + 0.922473i \(0.373835\pi\)
\(462\) 2.39925 0.111623
\(463\) 28.9063 1.34339 0.671694 0.740828i \(-0.265567\pi\)
0.671694 + 0.740828i \(0.265567\pi\)
\(464\) −19.7479 −0.916775
\(465\) 0 0
\(466\) 34.5522 1.60060
\(467\) 7.97738 0.369149 0.184574 0.982819i \(-0.440909\pi\)
0.184574 + 0.982819i \(0.440909\pi\)
\(468\) 4.51061 0.208503
\(469\) −1.15899 −0.0535172
\(470\) 0 0
\(471\) 8.47321 0.390425
\(472\) 34.9430 1.60838
\(473\) 10.9758 0.504667
\(474\) −22.8670 −1.05031
\(475\) 0 0
\(476\) 0.0179866 0.000824413 0
\(477\) −3.86284 −0.176867
\(478\) 6.82826 0.312317
\(479\) −2.05076 −0.0937014 −0.0468507 0.998902i \(-0.514919\pi\)
−0.0468507 + 0.998902i \(0.514919\pi\)
\(480\) 0 0
\(481\) −4.96093 −0.226199
\(482\) 1.99330 0.0907923
\(483\) 1.16287 0.0529124
\(484\) −0.336174 −0.0152807
\(485\) 0 0
\(486\) 63.6617 2.88775
\(487\) 19.7907 0.896804 0.448402 0.893832i \(-0.351993\pi\)
0.448402 + 0.893832i \(0.351993\pi\)
\(488\) −41.7586 −1.89032
\(489\) −47.3055 −2.13923
\(490\) 0 0
\(491\) −5.00033 −0.225662 −0.112831 0.993614i \(-0.535992\pi\)
−0.112831 + 0.993614i \(0.535992\pi\)
\(492\) −0.624002 −0.0281322
\(493\) −0.583242 −0.0262679
\(494\) −2.19239 −0.0986403
\(495\) 0 0
\(496\) −10.0064 −0.449301
\(497\) 1.72634 0.0774367
\(498\) 10.5316 0.471934
\(499\) 31.6046 1.41481 0.707407 0.706806i \(-0.249865\pi\)
0.707407 + 0.706806i \(0.249865\pi\)
\(500\) 0 0
\(501\) −56.0222 −2.50289
\(502\) −30.5923 −1.36540
\(503\) −19.0877 −0.851080 −0.425540 0.904940i \(-0.639916\pi\)
−0.425540 + 0.904940i \(0.639916\pi\)
\(504\) −13.4058 −0.597140
\(505\) 0 0
\(506\) 0.806423 0.0358499
\(507\) −33.3731 −1.48215
\(508\) 1.93891 0.0860251
\(509\) 42.8558 1.89955 0.949775 0.312934i \(-0.101312\pi\)
0.949775 + 0.312934i \(0.101312\pi\)
\(510\) 0 0
\(511\) −3.16328 −0.139935
\(512\) −25.4185 −1.12335
\(513\) 16.1539 0.713211
\(514\) −3.29275 −0.145237
\(515\) 0 0
\(516\) −12.1786 −0.536132
\(517\) 8.90408 0.391601
\(518\) 2.12167 0.0932210
\(519\) −82.4189 −3.61779
\(520\) 0 0
\(521\) −1.32945 −0.0582443 −0.0291222 0.999576i \(-0.509271\pi\)
−0.0291222 + 0.999576i \(0.509271\pi\)
\(522\) 62.5534 2.73789
\(523\) 19.3395 0.845656 0.422828 0.906210i \(-0.361037\pi\)
0.422828 + 0.906210i \(0.361037\pi\)
\(524\) 2.11758 0.0925069
\(525\) 0 0
\(526\) −5.24871 −0.228854
\(527\) −0.295532 −0.0128736
\(528\) −10.6103 −0.461756
\(529\) −22.6091 −0.983006
\(530\) 0 0
\(531\) −91.5395 −3.97248
\(532\) −0.189448 −0.00821361
\(533\) −0.955847 −0.0414023
\(534\) −59.6257 −2.58025
\(535\) 0 0
\(536\) 6.19746 0.267690
\(537\) 7.44563 0.321303
\(538\) 0.234430 0.0101070
\(539\) −6.68242 −0.287832
\(540\) 0 0
\(541\) −13.6158 −0.585390 −0.292695 0.956206i \(-0.594552\pi\)
−0.292695 + 0.956206i \(0.594552\pi\)
\(542\) −5.36147 −0.230295
\(543\) 16.9827 0.728798
\(544\) −0.178519 −0.00765393
\(545\) 0 0
\(546\) −4.07794 −0.174520
\(547\) −35.4126 −1.51413 −0.757066 0.653339i \(-0.773368\pi\)
−0.757066 + 0.653339i \(0.773368\pi\)
\(548\) −3.32802 −0.142166
\(549\) 109.394 4.66882
\(550\) 0 0
\(551\) 6.14313 0.261706
\(552\) −6.21821 −0.264665
\(553\) −3.02679 −0.128712
\(554\) 40.8052 1.73365
\(555\) 0 0
\(556\) −7.20702 −0.305646
\(557\) −2.52845 −0.107134 −0.0535669 0.998564i \(-0.517059\pi\)
−0.0535669 + 0.998564i \(0.517059\pi\)
\(558\) 31.6962 1.34181
\(559\) −18.6552 −0.789030
\(560\) 0 0
\(561\) −0.313369 −0.0132304
\(562\) 1.78459 0.0752784
\(563\) −22.8087 −0.961274 −0.480637 0.876920i \(-0.659594\pi\)
−0.480637 + 0.876920i \(0.659594\pi\)
\(564\) −9.87986 −0.416017
\(565\) 0 0
\(566\) 32.4948 1.36586
\(567\) 16.7008 0.701368
\(568\) −9.23122 −0.387334
\(569\) −14.2866 −0.598926 −0.299463 0.954108i \(-0.596807\pi\)
−0.299463 + 0.954108i \(0.596807\pi\)
\(570\) 0 0
\(571\) −11.3152 −0.473527 −0.236764 0.971567i \(-0.576087\pi\)
−0.236764 + 0.971567i \(0.576087\pi\)
\(572\) 0.571385 0.0238908
\(573\) −4.07598 −0.170277
\(574\) 0.408793 0.0170627
\(575\) 0 0
\(576\) 69.9002 2.91251
\(577\) −43.3748 −1.80572 −0.902858 0.429938i \(-0.858535\pi\)
−0.902858 + 0.429938i \(0.858535\pi\)
\(578\) −21.9166 −0.911609
\(579\) −21.0965 −0.876740
\(580\) 0 0
\(581\) 1.39402 0.0578338
\(582\) 26.1324 1.08322
\(583\) −0.489328 −0.0202659
\(584\) 16.9150 0.699946
\(585\) 0 0
\(586\) 15.2766 0.631072
\(587\) −21.8206 −0.900633 −0.450317 0.892869i \(-0.648689\pi\)
−0.450317 + 0.892869i \(0.648689\pi\)
\(588\) 7.41473 0.305779
\(589\) 3.11276 0.128259
\(590\) 0 0
\(591\) 17.2693 0.710365
\(592\) −9.38278 −0.385630
\(593\) −40.6744 −1.67030 −0.835149 0.550025i \(-0.814618\pi\)
−0.835149 + 0.550025i \(0.814618\pi\)
\(594\) 20.8368 0.854944
\(595\) 0 0
\(596\) 1.70056 0.0696575
\(597\) −63.0616 −2.58094
\(598\) −1.37065 −0.0560501
\(599\) −33.5105 −1.36920 −0.684601 0.728918i \(-0.740024\pi\)
−0.684601 + 0.728918i \(0.740024\pi\)
\(600\) 0 0
\(601\) 8.85454 0.361184 0.180592 0.983558i \(-0.442199\pi\)
0.180592 + 0.983558i \(0.442199\pi\)
\(602\) 7.97837 0.325174
\(603\) −16.2353 −0.661154
\(604\) 4.52309 0.184042
\(605\) 0 0
\(606\) 52.8680 2.14761
\(607\) −38.7846 −1.57422 −0.787109 0.616814i \(-0.788423\pi\)
−0.787109 + 0.616814i \(0.788423\pi\)
\(608\) 1.88029 0.0762559
\(609\) 11.4265 0.463024
\(610\) 0 0
\(611\) −15.1340 −0.612255
\(612\) 0.251959 0.0101848
\(613\) −11.4604 −0.462883 −0.231441 0.972849i \(-0.574344\pi\)
−0.231441 + 0.972849i \(0.574344\pi\)
\(614\) −16.1230 −0.650673
\(615\) 0 0
\(616\) −1.69818 −0.0684217
\(617\) 44.6630 1.79806 0.899032 0.437884i \(-0.144272\pi\)
0.899032 + 0.437884i \(0.144272\pi\)
\(618\) −45.5698 −1.83309
\(619\) −10.0641 −0.404508 −0.202254 0.979333i \(-0.564827\pi\)
−0.202254 + 0.979333i \(0.564827\pi\)
\(620\) 0 0
\(621\) 10.0992 0.405266
\(622\) −5.17320 −0.207427
\(623\) −7.89237 −0.316201
\(624\) 18.0341 0.721940
\(625\) 0 0
\(626\) 25.6001 1.02319
\(627\) 3.30063 0.131815
\(628\) −0.863009 −0.0344378
\(629\) −0.277114 −0.0110492
\(630\) 0 0
\(631\) −32.9805 −1.31293 −0.656466 0.754356i \(-0.727949\pi\)
−0.656466 + 0.754356i \(0.727949\pi\)
\(632\) 16.1851 0.643811
\(633\) −11.2887 −0.448684
\(634\) −20.9772 −0.833110
\(635\) 0 0
\(636\) 0.542952 0.0215295
\(637\) 11.3579 0.450016
\(638\) 7.92399 0.313714
\(639\) 24.1828 0.956658
\(640\) 0 0
\(641\) −15.4514 −0.610292 −0.305146 0.952306i \(-0.598705\pi\)
−0.305146 + 0.952306i \(0.598705\pi\)
\(642\) 10.7805 0.425474
\(643\) 26.4799 1.04427 0.522133 0.852864i \(-0.325136\pi\)
0.522133 + 0.852864i \(0.325136\pi\)
\(644\) −0.118440 −0.00466719
\(645\) 0 0
\(646\) −0.122465 −0.00481833
\(647\) 40.9548 1.61010 0.805050 0.593207i \(-0.202138\pi\)
0.805050 + 0.593207i \(0.202138\pi\)
\(648\) −89.3041 −3.50820
\(649\) −11.5958 −0.455176
\(650\) 0 0
\(651\) 5.78986 0.226923
\(652\) 4.81813 0.188693
\(653\) −23.3051 −0.911997 −0.455999 0.889980i \(-0.650718\pi\)
−0.455999 + 0.889980i \(0.650718\pi\)
\(654\) −43.7846 −1.71211
\(655\) 0 0
\(656\) −1.80782 −0.0705837
\(657\) −44.3118 −1.72877
\(658\) 6.47244 0.252322
\(659\) 10.8517 0.422723 0.211361 0.977408i \(-0.432210\pi\)
0.211361 + 0.977408i \(0.432210\pi\)
\(660\) 0 0
\(661\) 8.94285 0.347837 0.173918 0.984760i \(-0.444357\pi\)
0.173918 + 0.984760i \(0.444357\pi\)
\(662\) 22.5662 0.877062
\(663\) 0.532623 0.0206854
\(664\) −7.45425 −0.289281
\(665\) 0 0
\(666\) 29.7208 1.15166
\(667\) 3.84060 0.148709
\(668\) 5.70594 0.220769
\(669\) −35.0352 −1.35454
\(670\) 0 0
\(671\) 13.8576 0.534965
\(672\) 3.49742 0.134916
\(673\) −8.94979 −0.344989 −0.172494 0.985010i \(-0.555183\pi\)
−0.172494 + 0.985010i \(0.555183\pi\)
\(674\) −43.9125 −1.69145
\(675\) 0 0
\(676\) 3.39910 0.130735
\(677\) 8.30060 0.319018 0.159509 0.987196i \(-0.449009\pi\)
0.159509 + 0.987196i \(0.449009\pi\)
\(678\) −72.4333 −2.78178
\(679\) 3.45902 0.132745
\(680\) 0 0
\(681\) 23.7305 0.909353
\(682\) 4.01513 0.153747
\(683\) 37.6902 1.44217 0.721087 0.692844i \(-0.243643\pi\)
0.721087 + 0.692844i \(0.243643\pi\)
\(684\) −2.65382 −0.101471
\(685\) 0 0
\(686\) −9.94585 −0.379734
\(687\) −69.9604 −2.66916
\(688\) −35.2831 −1.34516
\(689\) 0.831695 0.0316851
\(690\) 0 0
\(691\) −49.0286 −1.86514 −0.932568 0.360994i \(-0.882437\pi\)
−0.932568 + 0.360994i \(0.882437\pi\)
\(692\) 8.39449 0.319111
\(693\) 4.44869 0.168992
\(694\) 24.2689 0.921234
\(695\) 0 0
\(696\) −61.1007 −2.31602
\(697\) −0.0533928 −0.00202240
\(698\) 34.6509 1.31155
\(699\) 88.4136 3.34411
\(700\) 0 0
\(701\) −8.98928 −0.339520 −0.169760 0.985485i \(-0.554299\pi\)
−0.169760 + 0.985485i \(0.554299\pi\)
\(702\) −35.4156 −1.33668
\(703\) 2.91877 0.110083
\(704\) 8.85465 0.333722
\(705\) 0 0
\(706\) −11.5094 −0.433163
\(707\) 6.99788 0.263182
\(708\) 12.8666 0.483556
\(709\) 22.4894 0.844607 0.422303 0.906455i \(-0.361222\pi\)
0.422303 + 0.906455i \(0.361222\pi\)
\(710\) 0 0
\(711\) −42.3999 −1.59012
\(712\) 42.2028 1.58162
\(713\) 1.94605 0.0728803
\(714\) −0.227790 −0.00852483
\(715\) 0 0
\(716\) −0.758349 −0.0283408
\(717\) 17.4724 0.652520
\(718\) 6.24142 0.232928
\(719\) −27.3255 −1.01907 −0.509534 0.860450i \(-0.670182\pi\)
−0.509534 + 0.860450i \(0.670182\pi\)
\(720\) 0 0
\(721\) −6.03186 −0.224638
\(722\) 1.28989 0.0480049
\(723\) 5.10054 0.189691
\(724\) −1.72971 −0.0642843
\(725\) 0 0
\(726\) 4.25747 0.158009
\(727\) 45.7801 1.69789 0.848945 0.528482i \(-0.177238\pi\)
0.848945 + 0.528482i \(0.177238\pi\)
\(728\) 2.88635 0.106975
\(729\) 73.9938 2.74051
\(730\) 0 0
\(731\) −1.04206 −0.0385421
\(732\) −15.3762 −0.568320
\(733\) 19.9594 0.737219 0.368610 0.929584i \(-0.379834\pi\)
0.368610 + 0.929584i \(0.379834\pi\)
\(734\) 22.9468 0.846982
\(735\) 0 0
\(736\) 1.17553 0.0433306
\(737\) −2.05662 −0.0757567
\(738\) 5.72644 0.210793
\(739\) 22.5149 0.828226 0.414113 0.910226i \(-0.364092\pi\)
0.414113 + 0.910226i \(0.364092\pi\)
\(740\) 0 0
\(741\) −5.60998 −0.206088
\(742\) −0.355696 −0.0130580
\(743\) 26.2969 0.964739 0.482370 0.875968i \(-0.339776\pi\)
0.482370 + 0.875968i \(0.339776\pi\)
\(744\) −30.9601 −1.13505
\(745\) 0 0
\(746\) 7.89906 0.289205
\(747\) 19.5277 0.714482
\(748\) 0.0319171 0.00116700
\(749\) 1.42697 0.0521404
\(750\) 0 0
\(751\) −27.3472 −0.997914 −0.498957 0.866627i \(-0.666283\pi\)
−0.498957 + 0.866627i \(0.666283\pi\)
\(752\) −28.6234 −1.04379
\(753\) −78.2809 −2.85272
\(754\) −13.4681 −0.490481
\(755\) 0 0
\(756\) −3.06032 −0.111303
\(757\) 23.2243 0.844102 0.422051 0.906572i \(-0.361310\pi\)
0.422051 + 0.906572i \(0.361310\pi\)
\(758\) −31.7036 −1.15153
\(759\) 2.06351 0.0749006
\(760\) 0 0
\(761\) −38.8028 −1.40660 −0.703300 0.710893i \(-0.748291\pi\)
−0.703300 + 0.710893i \(0.748291\pi\)
\(762\) −24.5552 −0.889542
\(763\) −5.79555 −0.209813
\(764\) 0.415145 0.0150194
\(765\) 0 0
\(766\) −37.8670 −1.36819
\(767\) 19.7090 0.711653
\(768\) −25.8356 −0.932261
\(769\) 2.29406 0.0827261 0.0413630 0.999144i \(-0.486830\pi\)
0.0413630 + 0.999144i \(0.486830\pi\)
\(770\) 0 0
\(771\) −8.42561 −0.303441
\(772\) 2.14871 0.0773337
\(773\) −52.2893 −1.88071 −0.940357 0.340189i \(-0.889509\pi\)
−0.940357 + 0.340189i \(0.889509\pi\)
\(774\) 111.762 4.01722
\(775\) 0 0
\(776\) −18.4964 −0.663982
\(777\) 5.42902 0.194765
\(778\) 30.1025 1.07923
\(779\) 0.562373 0.0201491
\(780\) 0 0
\(781\) 3.06337 0.109616
\(782\) −0.0765634 −0.00273790
\(783\) 99.2354 3.54638
\(784\) 21.4816 0.767199
\(785\) 0 0
\(786\) −26.8180 −0.956567
\(787\) −9.08106 −0.323705 −0.161852 0.986815i \(-0.551747\pi\)
−0.161852 + 0.986815i \(0.551747\pi\)
\(788\) −1.75891 −0.0626584
\(789\) −13.4306 −0.478142
\(790\) 0 0
\(791\) −9.58765 −0.340898
\(792\) −23.7884 −0.845285
\(793\) −23.5532 −0.836400
\(794\) −18.6641 −0.662363
\(795\) 0 0
\(796\) 6.42292 0.227654
\(797\) −16.3422 −0.578870 −0.289435 0.957198i \(-0.593467\pi\)
−0.289435 + 0.957198i \(0.593467\pi\)
\(798\) 2.39925 0.0849327
\(799\) −0.845371 −0.0299071
\(800\) 0 0
\(801\) −110.558 −3.90636
\(802\) −16.4710 −0.581612
\(803\) −5.61322 −0.198086
\(804\) 2.28200 0.0804801
\(805\) 0 0
\(806\) −6.82439 −0.240379
\(807\) 0.599868 0.0211164
\(808\) −37.4197 −1.31642
\(809\) −3.54010 −0.124463 −0.0622317 0.998062i \(-0.519822\pi\)
−0.0622317 + 0.998062i \(0.519822\pi\)
\(810\) 0 0
\(811\) −32.9605 −1.15740 −0.578699 0.815541i \(-0.696439\pi\)
−0.578699 + 0.815541i \(0.696439\pi\)
\(812\) −1.16380 −0.0408415
\(813\) −13.7192 −0.481152
\(814\) 3.76490 0.131960
\(815\) 0 0
\(816\) 1.00737 0.0352649
\(817\) 10.9758 0.383994
\(818\) −29.7903 −1.04159
\(819\) −7.56129 −0.264213
\(820\) 0 0
\(821\) 18.7416 0.654086 0.327043 0.945010i \(-0.393948\pi\)
0.327043 + 0.945010i \(0.393948\pi\)
\(822\) 42.1476 1.47007
\(823\) 22.5046 0.784462 0.392231 0.919867i \(-0.371703\pi\)
0.392231 + 0.919867i \(0.371703\pi\)
\(824\) 32.2541 1.12362
\(825\) 0 0
\(826\) −8.42909 −0.293286
\(827\) 11.9403 0.415205 0.207603 0.978213i \(-0.433434\pi\)
0.207603 + 0.978213i \(0.433434\pi\)
\(828\) −1.65913 −0.0576587
\(829\) 39.5586 1.37393 0.686964 0.726691i \(-0.258943\pi\)
0.686964 + 0.726691i \(0.258943\pi\)
\(830\) 0 0
\(831\) 104.414 3.62208
\(832\) −15.0500 −0.521764
\(833\) 0.634443 0.0219821
\(834\) 91.2729 3.16052
\(835\) 0 0
\(836\) −0.336174 −0.0116268
\(837\) 50.2832 1.73804
\(838\) −2.63977 −0.0911894
\(839\) −34.8294 −1.20244 −0.601222 0.799082i \(-0.705319\pi\)
−0.601222 + 0.799082i \(0.705319\pi\)
\(840\) 0 0
\(841\) 8.73807 0.301313
\(842\) 5.62454 0.193835
\(843\) 4.56648 0.157278
\(844\) 1.14977 0.0395766
\(845\) 0 0
\(846\) 90.6671 3.11720
\(847\) 0.563541 0.0193635
\(848\) 1.57301 0.0540175
\(849\) 83.1489 2.85366
\(850\) 0 0
\(851\) 1.82477 0.0625523
\(852\) −3.39908 −0.116451
\(853\) 55.1856 1.88952 0.944759 0.327765i \(-0.106295\pi\)
0.944759 + 0.327765i \(0.106295\pi\)
\(854\) 10.0732 0.344697
\(855\) 0 0
\(856\) −7.63043 −0.260803
\(857\) 21.4134 0.731468 0.365734 0.930719i \(-0.380818\pi\)
0.365734 + 0.930719i \(0.380818\pi\)
\(858\) −7.23628 −0.247042
\(859\) −36.5639 −1.24754 −0.623772 0.781606i \(-0.714401\pi\)
−0.623772 + 0.781606i \(0.714401\pi\)
\(860\) 0 0
\(861\) 1.04604 0.0356488
\(862\) 19.6066 0.667802
\(863\) 52.7117 1.79433 0.897163 0.441699i \(-0.145624\pi\)
0.897163 + 0.441699i \(0.145624\pi\)
\(864\) 30.3740 1.03334
\(865\) 0 0
\(866\) 18.7640 0.637627
\(867\) −56.0810 −1.90461
\(868\) −0.589706 −0.0200159
\(869\) −5.37103 −0.182200
\(870\) 0 0
\(871\) 3.49558 0.118443
\(872\) 30.9905 1.04947
\(873\) 48.4545 1.63994
\(874\) 0.806423 0.0272777
\(875\) 0 0
\(876\) 6.22836 0.210437
\(877\) 18.5720 0.627133 0.313566 0.949566i \(-0.398476\pi\)
0.313566 + 0.949566i \(0.398476\pi\)
\(878\) 44.2517 1.49342
\(879\) 39.0905 1.31849
\(880\) 0 0
\(881\) −32.2045 −1.08500 −0.542499 0.840056i \(-0.682522\pi\)
−0.542499 + 0.840056i \(0.682522\pi\)
\(882\) −68.0448 −2.29119
\(883\) −41.2458 −1.38803 −0.694016 0.719960i \(-0.744160\pi\)
−0.694016 + 0.719960i \(0.744160\pi\)
\(884\) −0.0542484 −0.00182457
\(885\) 0 0
\(886\) 2.01396 0.0676602
\(887\) −24.4058 −0.819468 −0.409734 0.912205i \(-0.634379\pi\)
−0.409734 + 0.912205i \(0.634379\pi\)
\(888\) −29.0306 −0.974203
\(889\) −3.25026 −0.109010
\(890\) 0 0
\(891\) 29.6355 0.992826
\(892\) 3.56839 0.119478
\(893\) 8.90408 0.297964
\(894\) −21.5366 −0.720293
\(895\) 0 0
\(896\) 4.31727 0.144230
\(897\) −3.50728 −0.117105
\(898\) −26.8193 −0.894970
\(899\) 19.1221 0.637758
\(900\) 0 0
\(901\) 0.0464578 0.00154773
\(902\) 0.725401 0.0241532
\(903\) 20.4154 0.679381
\(904\) 51.2680 1.70515
\(905\) 0 0
\(906\) −57.2825 −1.90308
\(907\) 47.4499 1.57555 0.787774 0.615965i \(-0.211234\pi\)
0.787774 + 0.615965i \(0.211234\pi\)
\(908\) −2.41698 −0.0802104
\(909\) 98.0276 3.25137
\(910\) 0 0
\(911\) 50.2159 1.66373 0.831863 0.554981i \(-0.187275\pi\)
0.831863 + 0.554981i \(0.187275\pi\)
\(912\) −10.6103 −0.351343
\(913\) 2.47369 0.0818671
\(914\) 3.88046 0.128354
\(915\) 0 0
\(916\) 7.12557 0.235435
\(917\) −3.54977 −0.117224
\(918\) −1.97829 −0.0652932
\(919\) −0.561735 −0.0185299 −0.00926497 0.999957i \(-0.502949\pi\)
−0.00926497 + 0.999957i \(0.502949\pi\)
\(920\) 0 0
\(921\) −41.2563 −1.35944
\(922\) 21.3841 0.704247
\(923\) −5.20672 −0.171381
\(924\) −0.625298 −0.0205708
\(925\) 0 0
\(926\) 37.2860 1.22529
\(927\) −84.4953 −2.77519
\(928\) 11.5509 0.379176
\(929\) 24.7806 0.813025 0.406513 0.913645i \(-0.366745\pi\)
0.406513 + 0.913645i \(0.366745\pi\)
\(930\) 0 0
\(931\) −6.68242 −0.219008
\(932\) −9.00506 −0.294971
\(933\) −13.2374 −0.433373
\(934\) 10.2900 0.336698
\(935\) 0 0
\(936\) 40.4324 1.32158
\(937\) 31.7278 1.03650 0.518251 0.855229i \(-0.326583\pi\)
0.518251 + 0.855229i \(0.326583\pi\)
\(938\) −1.49497 −0.0488127
\(939\) 65.5067 2.13773
\(940\) 0 0
\(941\) −24.1062 −0.785839 −0.392920 0.919573i \(-0.628535\pi\)
−0.392920 + 0.919573i \(0.628535\pi\)
\(942\) 10.9295 0.356104
\(943\) 0.351587 0.0114493
\(944\) 37.2764 1.21324
\(945\) 0 0
\(946\) 14.1576 0.460303
\(947\) −52.7844 −1.71526 −0.857631 0.514266i \(-0.828064\pi\)
−0.857631 + 0.514266i \(0.828064\pi\)
\(948\) 5.95963 0.193560
\(949\) 9.54061 0.309701
\(950\) 0 0
\(951\) −53.6773 −1.74060
\(952\) 0.161229 0.00522546
\(953\) −43.2711 −1.40169 −0.700845 0.713314i \(-0.747193\pi\)
−0.700845 + 0.713314i \(0.747193\pi\)
\(954\) −4.98265 −0.161319
\(955\) 0 0
\(956\) −1.77959 −0.0575562
\(957\) 20.2762 0.655437
\(958\) −2.64526 −0.0854644
\(959\) 5.57888 0.180151
\(960\) 0 0
\(961\) −21.3107 −0.687442
\(962\) −6.39908 −0.206315
\(963\) 19.9893 0.644145
\(964\) −0.519497 −0.0167319
\(965\) 0 0
\(966\) 1.49998 0.0482610
\(967\) −28.7928 −0.925915 −0.462958 0.886380i \(-0.653212\pi\)
−0.462958 + 0.886380i \(0.653212\pi\)
\(968\) −3.01342 −0.0968549
\(969\) −0.313369 −0.0100669
\(970\) 0 0
\(971\) 27.3569 0.877923 0.438962 0.898506i \(-0.355346\pi\)
0.438962 + 0.898506i \(0.355346\pi\)
\(972\) −16.5916 −0.532177
\(973\) 12.0814 0.387311
\(974\) 25.5280 0.817968
\(975\) 0 0
\(976\) −44.5470 −1.42592
\(977\) 48.1900 1.54174 0.770868 0.636995i \(-0.219823\pi\)
0.770868 + 0.636995i \(0.219823\pi\)
\(978\) −61.0191 −1.95117
\(979\) −14.0050 −0.447601
\(980\) 0 0
\(981\) −81.1852 −2.59204
\(982\) −6.44990 −0.205824
\(983\) −38.6466 −1.23264 −0.616318 0.787498i \(-0.711376\pi\)
−0.616318 + 0.787498i \(0.711376\pi\)
\(984\) −5.59346 −0.178313
\(985\) 0 0
\(986\) −0.752320 −0.0239587
\(987\) 16.5619 0.527172
\(988\) 0.571385 0.0181782
\(989\) 6.86189 0.218196
\(990\) 0 0
\(991\) 4.47927 0.142289 0.0711444 0.997466i \(-0.477335\pi\)
0.0711444 + 0.997466i \(0.477335\pi\)
\(992\) 5.85290 0.185830
\(993\) 57.7434 1.83243
\(994\) 2.22679 0.0706295
\(995\) 0 0
\(996\) −2.74477 −0.0869715
\(997\) 4.58855 0.145321 0.0726605 0.997357i \(-0.476851\pi\)
0.0726605 + 0.997357i \(0.476851\pi\)
\(998\) 40.7665 1.29044
\(999\) 47.1494 1.49174
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 5225.2.a.y.1.10 yes 15
5.4 even 2 5225.2.a.r.1.6 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5225.2.a.r.1.6 15 5.4 even 2
5225.2.a.y.1.10 yes 15 1.1 even 1 trivial