Properties

Label 5225.2.a.r.1.6
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $1$
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: \(1\)
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 1.6
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 \(-0.650735\pi\)
−0.456046 + 0.889956i \(0.650735\pi\)
\(3\) −3.30063 −1.90562 −0.952811 0.303565i \(-0.901823\pi\)
−0.952811 + 0.303565i \(0.901823\pi\)
\(4\) −0.336174 −0.168087
\(5\) 0 0
\(6\) 4.25747 1.73810
\(7\) −0.563541 −0.212998 −0.106499 0.994313i \(-0.533964\pi\)
−0.106499 + 0.994313i \(0.533964\pi\)
\(8\) 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 \(-0.424261\pi\)
0.235702 + 0.971825i \(0.424261\pi\)
\(14\) 0.726907 0.194274
\(15\) 0 0
\(16\) −3.21464 −0.803660
\(17\) 0.0949420 0.0230268 0.0115134 0.999934i \(-0.496335\pi\)
0.0115134 + 0.999934i \(0.496335\pi\)
\(18\) −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 \(-0.520762\pi\)
−0.0651801 + 0.997874i \(0.520762\pi\)
\(24\) −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 \(-0.577122\pi\)
−0.239921 + 0.970792i \(0.577122\pi\)
\(38\) −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 \(-0.815631\pi\)
−0.836895 + 0.547364i \(0.815631\pi\)
\(44\) −0.336174 −0.0506802
\(45\) 0 0
\(46\) 0.806423 0.118901
\(47\) −8.90408 −1.29879 −0.649396 0.760450i \(-0.724978\pi\)
−0.649396 + 0.760450i \(0.724978\pi\)
\(48\) 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 \(-0.489300\pi\)
0.0336072 + 0.999435i \(0.489300\pi\)
\(54\) 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 \(-0.459905\pi\)
0.125628 + 0.992077i \(0.459905\pi\)
\(68\) −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 \(-0.393461\pi\)
0.328489 + 0.944508i \(0.393461\pi\)
\(74\) 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 \(-0.543348\pi\)
−0.135761 + 0.990742i \(0.543348\pi\)
\(84\) −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 \(-0.600868\pi\)
−0.311610 + 0.950210i \(0.600868\pi\)
\(98\) 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 \(-0.323196\pi\)
0.527324 + 0.849665i \(0.323196\pi\)
\(104\) 5.12181 0.502234
\(105\) 0 0
\(106\) −0.631181 −0.0613057
\(107\) −2.53215 −0.244792 −0.122396 0.992481i \(-0.539058\pi\)
−0.122396 + 0.992481i \(0.539058\pi\)
\(108\) 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 \(-0.204708\pi\)
0.800236 + 0.599686i \(0.204708\pi\)
\(114\) 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 \(-0.417630\pi\)
0.255894 + 0.966705i \(0.417630\pi\)
\(128\) −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 \(-0.638986\pi\)
−0.422894 + 0.906179i \(0.638986\pi\)
\(138\) −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 \(-0.532665\pi\)
−0.102440 + 0.994739i \(0.532665\pi\)
\(158\) 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 \(-0.310304\pi\)
0.561294 + 0.827616i \(0.310304\pi\)
\(164\) −0.189055 −0.0147627
\(165\) 0 0
\(166\) 3.19079 0.247654
\(167\) 16.9732 1.31342 0.656711 0.754142i \(-0.271947\pi\)
0.656711 + 0.754142i \(0.271947\pi\)
\(168\) 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 \(-0.101853\pi\)
0.949241 + 0.314549i \(0.101853\pi\)
\(174\) 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 \(-0.426114\pi\)
0.230041 + 0.973181i \(0.426114\pi\)
\(194\) 7.91738 0.568435
\(195\) 0 0
\(196\) 2.24646 0.160461
\(197\) −5.23212 −0.372773 −0.186387 0.982476i \(-0.559678\pi\)
−0.186387 + 0.982476i \(0.559678\pi\)
\(198\) −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 \(-0.384343\pi\)
0.355406 + 0.934712i \(0.384343\pi\)
\(224\) 1.05962 0.0707989
\(225\) 0 0
\(226\) −21.9453 −1.45978
\(227\) −7.18967 −0.477195 −0.238598 0.971119i \(-0.576688\pi\)
−0.238598 + 0.971119i \(0.576688\pi\)
\(228\) 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 \(-0.840746\pi\)
−0.877433 + 0.479699i \(0.840746\pi\)
\(234\) −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 \(-0.474630\pi\)
0.0796174 + 0.996825i \(0.474630\pi\)
\(258\) −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 \(-0.459961\pi\)
0.125456 + 0.992099i \(0.459961\pi\)
\(264\) −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 \(-0.899294\pi\)
−0.950368 + 0.311127i \(0.899294\pi\)
\(278\) −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 \(-0.769345\pi\)
−0.748749 + 0.662854i \(0.769345\pi\)
\(284\) −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 \(-0.612442\pi\)
−0.345947 + 0.938254i \(0.612442\pi\)
\(294\) −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 \(-0.383904\pi\)
0.356692 + 0.934222i \(0.383904\pi\)
\(308\) 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 \(-0.689545\pi\)
−0.560901 + 0.827883i \(0.689545\pi\)
\(314\) 3.31135 0.186870
\(315\) 0 0
\(316\) 1.80560 0.101573
\(317\) 16.2627 0.913406 0.456703 0.889619i \(-0.349030\pi\)
0.456703 + 0.889619i \(0.349030\pi\)
\(318\) 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 \(-0.122179\pi\)
0.927234 + 0.374482i \(0.122179\pi\)
\(338\) 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 \(-0.668512\pi\)
−0.505011 + 0.863113i \(0.668512\pi\)
\(348\) 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 \(-0.423687\pi\)
0.237456 + 0.971398i \(0.423687\pi\)
\(354\) −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 \(-0.653697\pi\)
−0.464307 + 0.885674i \(0.653697\pi\)
\(368\) 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 \(-0.550678\pi\)
−0.158539 + 0.987353i \(0.550678\pi\)
\(374\) −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 \(-0.230040\pi\)
0.750029 + 0.661405i \(0.230040\pi\)
\(384\) 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 \(-0.381718\pi\)
0.363101 + 0.931750i \(0.381718\pi\)
\(398\) 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 \(-0.613662\pi\)
−0.349541 + 0.936921i \(0.613662\pi\)
\(434\) 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 \(-0.511809\pi\)
−0.0370907 + 0.999312i \(0.511809\pi\)
\(444\) −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 \(-0.522416\pi\)
−0.0703624 + 0.997521i \(0.522416\pi\)
\(458\) 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 \(-0.734433\pi\)
−0.671694 + 0.740828i \(0.734433\pi\)
\(464\) −19.7479 −0.916775
\(465\) 0 0
\(466\) 34.5522 1.60060
\(467\) −7.97738 −0.369149 −0.184574 0.982819i \(-0.559091\pi\)
−0.184574 + 0.982819i \(0.559091\pi\)
\(468\) −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 \(-0.648007\pi\)
−0.448402 + 0.893832i \(0.648007\pi\)
\(488\) 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 \(-0.360084\pi\)
0.425540 + 0.904940i \(0.360084\pi\)
\(504\) −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 \(-0.638963\pi\)
−0.422828 + 0.906210i \(0.638963\pi\)
\(524\) 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 \(-0.226632\pi\)
0.757066 + 0.653339i \(0.226632\pi\)
\(548\) 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 \(-0.482941\pi\)
0.0535669 + 0.998564i \(0.482941\pi\)
\(558\) −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 \(-0.340406\pi\)
0.480637 + 0.876920i \(0.340406\pi\)
\(564\) −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 \(-0.141465\pi\)
0.902858 + 0.429938i \(0.141465\pi\)
\(578\) 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 \(-0.351311\pi\)
0.450317 + 0.892869i \(0.351311\pi\)
\(588\) −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 \(-0.185382\pi\)
0.835149 + 0.550025i \(0.185382\pi\)
\(594\) 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 \(-0.211577\pi\)
0.787109 + 0.616814i \(0.211577\pi\)
\(608\) −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 \(-0.425656\pi\)
0.231441 + 0.972849i \(0.425656\pi\)
\(614\) −16.1230 −0.650673
\(615\) 0 0
\(616\) −1.69818 −0.0684217
\(617\) −44.6630 −1.79806 −0.899032 0.437884i \(-0.855728\pi\)
−0.899032 + 0.437884i \(0.855728\pi\)
\(618\) 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 \(-0.674864\pi\)
−0.522133 + 0.852864i \(0.674864\pi\)
\(644\) −0.118440 −0.00466719
\(645\) 0 0
\(646\) −0.122465 −0.00481833
\(647\) −40.9548 −1.61010 −0.805050 0.593207i \(-0.797862\pi\)
−0.805050 + 0.593207i \(0.797862\pi\)
\(648\) 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 \(-0.349282\pi\)
0.455999 + 0.889980i \(0.349282\pi\)
\(654\) −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 \(-0.444817\pi\)
0.172494 + 0.985010i \(0.444817\pi\)
\(674\) −43.9125 −1.69145
\(675\) 0 0
\(676\) 3.39910 0.130735
\(677\) −8.30060 −0.319018 −0.159509 0.987196i \(-0.550991\pi\)
−0.159509 + 0.987196i \(0.550991\pi\)
\(678\) 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 \(-0.756357\pi\)
−0.721087 + 0.692844i \(0.756357\pi\)
\(684\) −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 \(-0.822762\pi\)
−0.848945 + 0.528482i \(0.822762\pi\)
\(728\) −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 \(-0.620166\pi\)
−0.368610 + 0.929584i \(0.620166\pi\)
\(734\) 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 \(-0.660224\pi\)
−0.482370 + 0.875968i \(0.660224\pi\)
\(744\) −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 \(-0.638690\pi\)
−0.422051 + 0.906572i \(0.638690\pi\)
\(758\) 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 \(-0.110491\pi\)
0.940357 + 0.340189i \(0.110491\pi\)
\(774\) 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 \(-0.448253\pi\)
0.161852 + 0.986815i \(0.448253\pi\)
\(788\) 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 \(-0.406533\pi\)
0.289435 + 0.957198i \(0.406533\pi\)
\(798\) −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 \(-0.628297\pi\)
−0.392231 + 0.919867i \(0.628297\pi\)
\(824\) 32.2541 1.12362
\(825\) 0 0
\(826\) −8.42909 −0.293286
\(827\) −11.9403 −0.415205 −0.207603 0.978213i \(-0.566566\pi\)
−0.207603 + 0.978213i \(0.566566\pi\)
\(828\) 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 \(-0.893705\pi\)
−0.944759 + 0.327765i \(0.893705\pi\)
\(854\) 10.0732 0.344697
\(855\) 0 0
\(856\) −7.63043 −0.260803
\(857\) −21.4134 −0.731468 −0.365734 0.930719i \(-0.619182\pi\)
−0.365734 + 0.930719i \(0.619182\pi\)
\(858\) 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 \(-0.854376\pi\)
−0.897163 + 0.441699i \(0.854376\pi\)
\(864\) 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 \(-0.601524\pi\)
−0.313566 + 0.949566i \(0.601524\pi\)
\(878\) −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 \(-0.255840\pi\)
0.694016 + 0.719960i \(0.255840\pi\)
\(884\) −0.0542484 −0.00182457
\(885\) 0 0
\(886\) 2.01396 0.0676602
\(887\) 24.4058 0.819468 0.409734 0.912205i \(-0.365621\pi\)
0.409734 + 0.912205i \(0.365621\pi\)
\(888\) 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 \(-0.788766\pi\)
−0.787774 + 0.615965i \(0.788766\pi\)
\(908\) 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 \(-0.673417\pi\)
−0.518251 + 0.855229i \(0.673417\pi\)
\(938\) 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 \(-0.171936\pi\)
0.857631 + 0.514266i \(0.171936\pi\)
\(948\) −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 \(-0.252807\pi\)
0.700845 + 0.713314i \(0.252807\pi\)
\(954\) −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 \(-0.346788\pi\)
0.462958 + 0.886380i \(0.346788\pi\)
\(968\) 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 \(-0.780177\pi\)
−0.770868 + 0.636995i \(0.780177\pi\)
\(978\) 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 \(-0.288624\pi\)
0.616318 + 0.787498i \(0.288624\pi\)
\(984\) −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 \(-0.523149\pi\)
−0.0726605 + 0.997357i \(0.523149\pi\)
\(998\) −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.r.1.6 15
5.4 even 2 5225.2.a.y.1.10 yes 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5225.2.a.r.1.6 15 1.1 even 1 trivial
5225.2.a.y.1.10 yes 15 5.4 even 2