Properties

Label 1045.2.a.h.1.3
Level $1045$
Weight $2$
Character 1045.1
Self dual yes
Analytic conductor $8.344$
Analytic rank $0$
Dimension $7$
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] = [1045,2,Mod(1,1045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, 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("1045.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1045.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: \(8.34436701122\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 10x^{5} + 8x^{4} + 27x^{3} - 16x^{2} - 18x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.08185\) of defining polynomial
Character \(\chi\) \(=\) 1045.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.08185 q^{2} -0.870874 q^{3} -0.829591 q^{4} -1.00000 q^{5} +0.942160 q^{6} +4.02863 q^{7} +3.06121 q^{8} -2.24158 q^{9} +O(q^{10})\) \(q-1.08185 q^{2} -0.870874 q^{3} -0.829591 q^{4} -1.00000 q^{5} +0.942160 q^{6} +4.02863 q^{7} +3.06121 q^{8} -2.24158 q^{9} +1.08185 q^{10} +1.00000 q^{11} +0.722469 q^{12} -3.57009 q^{13} -4.35839 q^{14} +0.870874 q^{15} -1.65260 q^{16} +2.13712 q^{17} +2.42506 q^{18} -1.00000 q^{19} +0.829591 q^{20} -3.50843 q^{21} -1.08185 q^{22} -1.52642 q^{23} -2.66593 q^{24} +1.00000 q^{25} +3.86231 q^{26} +4.56476 q^{27} -3.34212 q^{28} +0.640237 q^{29} -0.942160 q^{30} -2.79975 q^{31} -4.33454 q^{32} -0.870874 q^{33} -2.31206 q^{34} -4.02863 q^{35} +1.85959 q^{36} +6.88152 q^{37} +1.08185 q^{38} +3.10910 q^{39} -3.06121 q^{40} +2.11443 q^{41} +3.79561 q^{42} -11.2566 q^{43} -0.829591 q^{44} +2.24158 q^{45} +1.65136 q^{46} -5.49461 q^{47} +1.43921 q^{48} +9.22988 q^{49} -1.08185 q^{50} -1.86117 q^{51} +2.96171 q^{52} +10.3553 q^{53} -4.93840 q^{54} -1.00000 q^{55} +12.3325 q^{56} +0.870874 q^{57} -0.692643 q^{58} +10.5403 q^{59} -0.722469 q^{60} +7.73205 q^{61} +3.02892 q^{62} -9.03049 q^{63} +7.99454 q^{64} +3.57009 q^{65} +0.942160 q^{66} +7.97406 q^{67} -1.77294 q^{68} +1.32932 q^{69} +4.35839 q^{70} +7.60336 q^{71} -6.86193 q^{72} +3.48422 q^{73} -7.44481 q^{74} -0.870874 q^{75} +0.829591 q^{76} +4.02863 q^{77} -3.36359 q^{78} +5.50316 q^{79} +1.65260 q^{80} +2.74940 q^{81} -2.28750 q^{82} +15.4236 q^{83} +2.91056 q^{84} -2.13712 q^{85} +12.1781 q^{86} -0.557566 q^{87} +3.06121 q^{88} +17.3913 q^{89} -2.42506 q^{90} -14.3826 q^{91} +1.26630 q^{92} +2.43823 q^{93} +5.94436 q^{94} +1.00000 q^{95} +3.77484 q^{96} -6.28029 q^{97} -9.98538 q^{98} -2.24158 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} + 3 q^{3} + 7 q^{4} - 7 q^{5} + 8 q^{6} - q^{7} + 3 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + q^{2} + 3 q^{3} + 7 q^{4} - 7 q^{5} + 8 q^{6} - q^{7} + 3 q^{8} + 2 q^{9} - q^{10} + 7 q^{11} + 13 q^{12} + q^{13} + 12 q^{14} - 3 q^{15} + 3 q^{16} + q^{17} + 7 q^{18} - 7 q^{19} - 7 q^{20} + 5 q^{21} + q^{22} - 8 q^{23} + 25 q^{24} + 7 q^{25} + 12 q^{27} + 4 q^{28} + 11 q^{29} - 8 q^{30} + 7 q^{31} + 12 q^{32} + 3 q^{33} - 14 q^{34} + q^{35} + 7 q^{36} - 17 q^{37} - q^{38} + 30 q^{39} - 3 q^{40} + 17 q^{41} + 33 q^{42} - 3 q^{43} + 7 q^{44} - 2 q^{45} + 18 q^{46} + 14 q^{47} - 12 q^{48} + 6 q^{49} + q^{50} + 8 q^{51} - 17 q^{52} + 7 q^{53} - 27 q^{54} - 7 q^{55} + 36 q^{56} - 3 q^{57} - 15 q^{58} + 35 q^{59} - 13 q^{60} + 17 q^{61} + 46 q^{62} - 22 q^{63} + 5 q^{64} - q^{65} + 8 q^{66} + 4 q^{67} - 35 q^{68} - 4 q^{69} - 12 q^{70} + 10 q^{71} + 12 q^{72} + 22 q^{73} - 11 q^{74} + 3 q^{75} - 7 q^{76} - q^{77} - 41 q^{78} + 11 q^{79} - 3 q^{80} - 21 q^{81} - 14 q^{82} + 39 q^{83} + 21 q^{84} - q^{85} - 24 q^{86} - 2 q^{87} + 3 q^{88} + 18 q^{89} - 7 q^{90} - 22 q^{91} - 51 q^{92} + 10 q^{93} + 14 q^{94} + 7 q^{95} - 11 q^{96} - 4 q^{97} - 26 q^{98} + 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.08185 −0.764987 −0.382493 0.923958i \(-0.624935\pi\)
−0.382493 + 0.923958i \(0.624935\pi\)
\(3\) −0.870874 −0.502800 −0.251400 0.967883i \(-0.580891\pi\)
−0.251400 + 0.967883i \(0.580891\pi\)
\(4\) −0.829591 −0.414795
\(5\) −1.00000 −0.447214
\(6\) 0.942160 0.384635
\(7\) 4.02863 1.52268 0.761340 0.648353i \(-0.224542\pi\)
0.761340 + 0.648353i \(0.224542\pi\)
\(8\) 3.06121 1.08230
\(9\) −2.24158 −0.747193
\(10\) 1.08185 0.342112
\(11\) 1.00000 0.301511
\(12\) 0.722469 0.208559
\(13\) −3.57009 −0.990164 −0.495082 0.868846i \(-0.664862\pi\)
−0.495082 + 0.868846i \(0.664862\pi\)
\(14\) −4.35839 −1.16483
\(15\) 0.870874 0.224859
\(16\) −1.65260 −0.413149
\(17\) 2.13712 0.518328 0.259164 0.965833i \(-0.416553\pi\)
0.259164 + 0.965833i \(0.416553\pi\)
\(18\) 2.42506 0.571592
\(19\) −1.00000 −0.229416
\(20\) 0.829591 0.185502
\(21\) −3.50843 −0.765603
\(22\) −1.08185 −0.230652
\(23\) −1.52642 −0.318280 −0.159140 0.987256i \(-0.550872\pi\)
−0.159140 + 0.987256i \(0.550872\pi\)
\(24\) −2.66593 −0.544180
\(25\) 1.00000 0.200000
\(26\) 3.86231 0.757462
\(27\) 4.56476 0.878488
\(28\) −3.34212 −0.631600
\(29\) 0.640237 0.118889 0.0594445 0.998232i \(-0.481067\pi\)
0.0594445 + 0.998232i \(0.481067\pi\)
\(30\) −0.942160 −0.172014
\(31\) −2.79975 −0.502850 −0.251425 0.967877i \(-0.580899\pi\)
−0.251425 + 0.967877i \(0.580899\pi\)
\(32\) −4.33454 −0.766246
\(33\) −0.870874 −0.151600
\(34\) −2.31206 −0.396514
\(35\) −4.02863 −0.680963
\(36\) 1.85959 0.309932
\(37\) 6.88152 1.13132 0.565658 0.824640i \(-0.308622\pi\)
0.565658 + 0.824640i \(0.308622\pi\)
\(38\) 1.08185 0.175500
\(39\) 3.10910 0.497854
\(40\) −3.06121 −0.484019
\(41\) 2.11443 0.330218 0.165109 0.986275i \(-0.447202\pi\)
0.165109 + 0.986275i \(0.447202\pi\)
\(42\) 3.79561 0.585676
\(43\) −11.2566 −1.71662 −0.858311 0.513129i \(-0.828486\pi\)
−0.858311 + 0.513129i \(0.828486\pi\)
\(44\) −0.829591 −0.125066
\(45\) 2.24158 0.334155
\(46\) 1.65136 0.243480
\(47\) −5.49461 −0.801471 −0.400735 0.916194i \(-0.631245\pi\)
−0.400735 + 0.916194i \(0.631245\pi\)
\(48\) 1.43921 0.207731
\(49\) 9.22988 1.31855
\(50\) −1.08185 −0.152997
\(51\) −1.86117 −0.260615
\(52\) 2.96171 0.410715
\(53\) 10.3553 1.42241 0.711203 0.702987i \(-0.248151\pi\)
0.711203 + 0.702987i \(0.248151\pi\)
\(54\) −4.93840 −0.672031
\(55\) −1.00000 −0.134840
\(56\) 12.3325 1.64800
\(57\) 0.870874 0.115350
\(58\) −0.692643 −0.0909485
\(59\) 10.5403 1.37224 0.686118 0.727490i \(-0.259313\pi\)
0.686118 + 0.727490i \(0.259313\pi\)
\(60\) −0.722469 −0.0932704
\(61\) 7.73205 0.989987 0.494994 0.868897i \(-0.335170\pi\)
0.494994 + 0.868897i \(0.335170\pi\)
\(62\) 3.02892 0.384674
\(63\) −9.03049 −1.13773
\(64\) 7.99454 0.999317
\(65\) 3.57009 0.442815
\(66\) 0.942160 0.115972
\(67\) 7.97406 0.974187 0.487094 0.873350i \(-0.338057\pi\)
0.487094 + 0.873350i \(0.338057\pi\)
\(68\) −1.77294 −0.215000
\(69\) 1.32932 0.160031
\(70\) 4.35839 0.520928
\(71\) 7.60336 0.902353 0.451176 0.892435i \(-0.351004\pi\)
0.451176 + 0.892435i \(0.351004\pi\)
\(72\) −6.86193 −0.808686
\(73\) 3.48422 0.407797 0.203899 0.978992i \(-0.434639\pi\)
0.203899 + 0.978992i \(0.434639\pi\)
\(74\) −7.44481 −0.865441
\(75\) −0.870874 −0.100560
\(76\) 0.829591 0.0951606
\(77\) 4.02863 0.459105
\(78\) −3.36359 −0.380852
\(79\) 5.50316 0.619154 0.309577 0.950874i \(-0.399813\pi\)
0.309577 + 0.950874i \(0.399813\pi\)
\(80\) 1.65260 0.184766
\(81\) 2.74940 0.305489
\(82\) −2.28750 −0.252613
\(83\) 15.4236 1.69296 0.846479 0.532422i \(-0.178718\pi\)
0.846479 + 0.532422i \(0.178718\pi\)
\(84\) 2.91056 0.317568
\(85\) −2.13712 −0.231803
\(86\) 12.1781 1.31319
\(87\) −0.557566 −0.0597773
\(88\) 3.06121 0.326326
\(89\) 17.3913 1.84347 0.921735 0.387820i \(-0.126772\pi\)
0.921735 + 0.387820i \(0.126772\pi\)
\(90\) −2.42506 −0.255624
\(91\) −14.3826 −1.50770
\(92\) 1.26630 0.132021
\(93\) 2.43823 0.252833
\(94\) 5.94436 0.613114
\(95\) 1.00000 0.102598
\(96\) 3.77484 0.385268
\(97\) −6.28029 −0.637666 −0.318833 0.947811i \(-0.603291\pi\)
−0.318833 + 0.947811i \(0.603291\pi\)
\(98\) −9.98538 −1.00868
\(99\) −2.24158 −0.225287
\(100\) −0.829591 −0.0829591
\(101\) 9.34301 0.929664 0.464832 0.885399i \(-0.346115\pi\)
0.464832 + 0.885399i \(0.346115\pi\)
\(102\) 2.01351 0.199367
\(103\) −14.5265 −1.43133 −0.715667 0.698442i \(-0.753877\pi\)
−0.715667 + 0.698442i \(0.753877\pi\)
\(104\) −10.9288 −1.07165
\(105\) 3.50843 0.342388
\(106\) −11.2029 −1.08812
\(107\) 11.4273 1.10472 0.552361 0.833605i \(-0.313727\pi\)
0.552361 + 0.833605i \(0.313727\pi\)
\(108\) −3.78688 −0.364393
\(109\) 1.86373 0.178513 0.0892564 0.996009i \(-0.471551\pi\)
0.0892564 + 0.996009i \(0.471551\pi\)
\(110\) 1.08185 0.103151
\(111\) −5.99294 −0.568825
\(112\) −6.65771 −0.629094
\(113\) −11.8891 −1.11843 −0.559214 0.829023i \(-0.688897\pi\)
−0.559214 + 0.829023i \(0.688897\pi\)
\(114\) −0.942160 −0.0882413
\(115\) 1.52642 0.142339
\(116\) −0.531134 −0.0493146
\(117\) 8.00263 0.739843
\(118\) −11.4031 −1.04974
\(119\) 8.60968 0.789248
\(120\) 2.66593 0.243365
\(121\) 1.00000 0.0909091
\(122\) −8.36495 −0.757327
\(123\) −1.84140 −0.166034
\(124\) 2.32265 0.208580
\(125\) −1.00000 −0.0894427
\(126\) 9.76968 0.870352
\(127\) −10.5195 −0.933457 −0.466729 0.884401i \(-0.654567\pi\)
−0.466729 + 0.884401i \(0.654567\pi\)
\(128\) 0.0201534 0.00178133
\(129\) 9.80313 0.863117
\(130\) −3.86231 −0.338747
\(131\) −4.29351 −0.375126 −0.187563 0.982253i \(-0.560059\pi\)
−0.187563 + 0.982253i \(0.560059\pi\)
\(132\) 0.722469 0.0628829
\(133\) −4.02863 −0.349327
\(134\) −8.62678 −0.745240
\(135\) −4.56476 −0.392872
\(136\) 6.54217 0.560987
\(137\) −14.8142 −1.26566 −0.632831 0.774290i \(-0.718107\pi\)
−0.632831 + 0.774290i \(0.718107\pi\)
\(138\) −1.43813 −0.122421
\(139\) 0.911248 0.0772910 0.0386455 0.999253i \(-0.487696\pi\)
0.0386455 + 0.999253i \(0.487696\pi\)
\(140\) 3.34212 0.282460
\(141\) 4.78511 0.402979
\(142\) −8.22573 −0.690288
\(143\) −3.57009 −0.298546
\(144\) 3.70443 0.308702
\(145\) −0.640237 −0.0531688
\(146\) −3.76942 −0.311959
\(147\) −8.03806 −0.662968
\(148\) −5.70885 −0.469264
\(149\) −2.83838 −0.232529 −0.116264 0.993218i \(-0.537092\pi\)
−0.116264 + 0.993218i \(0.537092\pi\)
\(150\) 0.942160 0.0769270
\(151\) −0.0486666 −0.00396043 −0.00198022 0.999998i \(-0.500630\pi\)
−0.00198022 + 0.999998i \(0.500630\pi\)
\(152\) −3.06121 −0.248297
\(153\) −4.79053 −0.387291
\(154\) −4.35839 −0.351209
\(155\) 2.79975 0.224881
\(156\) −2.57928 −0.206508
\(157\) 4.43937 0.354301 0.177150 0.984184i \(-0.443312\pi\)
0.177150 + 0.984184i \(0.443312\pi\)
\(158\) −5.95362 −0.473644
\(159\) −9.01814 −0.715185
\(160\) 4.33454 0.342676
\(161\) −6.14937 −0.484638
\(162\) −2.97445 −0.233695
\(163\) 19.4777 1.52561 0.762807 0.646626i \(-0.223820\pi\)
0.762807 + 0.646626i \(0.223820\pi\)
\(164\) −1.75411 −0.136973
\(165\) 0.870874 0.0677975
\(166\) −16.6861 −1.29509
\(167\) 10.8523 0.839778 0.419889 0.907575i \(-0.362069\pi\)
0.419889 + 0.907575i \(0.362069\pi\)
\(168\) −10.7400 −0.828612
\(169\) −0.254480 −0.0195753
\(170\) 2.31206 0.177327
\(171\) 2.24158 0.171418
\(172\) 9.33841 0.712047
\(173\) −1.46585 −0.111447 −0.0557234 0.998446i \(-0.517746\pi\)
−0.0557234 + 0.998446i \(0.517746\pi\)
\(174\) 0.603205 0.0457289
\(175\) 4.02863 0.304536
\(176\) −1.65260 −0.124569
\(177\) −9.17932 −0.689960
\(178\) −18.8148 −1.41023
\(179\) 17.5848 1.31435 0.657175 0.753738i \(-0.271751\pi\)
0.657175 + 0.753738i \(0.271751\pi\)
\(180\) −1.85959 −0.138606
\(181\) 1.09957 0.0817307 0.0408654 0.999165i \(-0.486989\pi\)
0.0408654 + 0.999165i \(0.486989\pi\)
\(182\) 15.5598 1.15337
\(183\) −6.73364 −0.497765
\(184\) −4.67267 −0.344474
\(185\) −6.88152 −0.505940
\(186\) −2.63781 −0.193414
\(187\) 2.13712 0.156282
\(188\) 4.55827 0.332446
\(189\) 18.3897 1.33766
\(190\) −1.08185 −0.0784860
\(191\) 20.0312 1.44941 0.724703 0.689061i \(-0.241977\pi\)
0.724703 + 0.689061i \(0.241977\pi\)
\(192\) −6.96224 −0.502456
\(193\) −22.7060 −1.63441 −0.817205 0.576347i \(-0.804478\pi\)
−0.817205 + 0.576347i \(0.804478\pi\)
\(194\) 6.79436 0.487806
\(195\) −3.10910 −0.222647
\(196\) −7.65702 −0.546930
\(197\) −0.862697 −0.0614646 −0.0307323 0.999528i \(-0.509784\pi\)
−0.0307323 + 0.999528i \(0.509784\pi\)
\(198\) 2.42506 0.172342
\(199\) 18.2787 1.29574 0.647872 0.761749i \(-0.275659\pi\)
0.647872 + 0.761749i \(0.275659\pi\)
\(200\) 3.06121 0.216460
\(201\) −6.94441 −0.489821
\(202\) −10.1078 −0.711181
\(203\) 2.57928 0.181030
\(204\) 1.54401 0.108102
\(205\) −2.11443 −0.147678
\(206\) 15.7155 1.09495
\(207\) 3.42158 0.237816
\(208\) 5.89992 0.409086
\(209\) −1.00000 −0.0691714
\(210\) −3.79561 −0.261922
\(211\) 4.69164 0.322986 0.161493 0.986874i \(-0.448369\pi\)
0.161493 + 0.986874i \(0.448369\pi\)
\(212\) −8.59063 −0.590007
\(213\) −6.62157 −0.453703
\(214\) −12.3627 −0.845098
\(215\) 11.2566 0.767697
\(216\) 13.9737 0.950787
\(217\) −11.2792 −0.765680
\(218\) −2.01628 −0.136560
\(219\) −3.03432 −0.205040
\(220\) 0.829591 0.0559310
\(221\) −7.62971 −0.513230
\(222\) 6.48349 0.435144
\(223\) 0.856060 0.0573260 0.0286630 0.999589i \(-0.490875\pi\)
0.0286630 + 0.999589i \(0.490875\pi\)
\(224\) −17.4623 −1.16675
\(225\) −2.24158 −0.149439
\(226\) 12.8622 0.855583
\(227\) 4.53380 0.300919 0.150459 0.988616i \(-0.451925\pi\)
0.150459 + 0.988616i \(0.451925\pi\)
\(228\) −0.722469 −0.0478467
\(229\) −28.5069 −1.88379 −0.941896 0.335905i \(-0.890958\pi\)
−0.941896 + 0.335905i \(0.890958\pi\)
\(230\) −1.65136 −0.108887
\(231\) −3.50843 −0.230838
\(232\) 1.95990 0.128674
\(233\) 5.61324 0.367736 0.183868 0.982951i \(-0.441138\pi\)
0.183868 + 0.982951i \(0.441138\pi\)
\(234\) −8.65768 −0.565970
\(235\) 5.49461 0.358429
\(236\) −8.74418 −0.569197
\(237\) −4.79256 −0.311310
\(238\) −9.31442 −0.603764
\(239\) −20.7190 −1.34020 −0.670101 0.742270i \(-0.733749\pi\)
−0.670101 + 0.742270i \(0.733749\pi\)
\(240\) −1.43921 −0.0929003
\(241\) 11.4250 0.735952 0.367976 0.929835i \(-0.380051\pi\)
0.367976 + 0.929835i \(0.380051\pi\)
\(242\) −1.08185 −0.0695442
\(243\) −16.0887 −1.03209
\(244\) −6.41443 −0.410642
\(245\) −9.22988 −0.589675
\(246\) 1.99213 0.127013
\(247\) 3.57009 0.227159
\(248\) −8.57061 −0.544234
\(249\) −13.4320 −0.851219
\(250\) 1.08185 0.0684225
\(251\) −10.3880 −0.655685 −0.327843 0.944732i \(-0.606322\pi\)
−0.327843 + 0.944732i \(0.606322\pi\)
\(252\) 7.49161 0.471927
\(253\) −1.52642 −0.0959649
\(254\) 11.3806 0.714082
\(255\) 1.86117 0.116551
\(256\) −16.0109 −1.00068
\(257\) 4.53951 0.283167 0.141583 0.989926i \(-0.454781\pi\)
0.141583 + 0.989926i \(0.454781\pi\)
\(258\) −10.6056 −0.660273
\(259\) 27.7231 1.72263
\(260\) −2.96171 −0.183678
\(261\) −1.43514 −0.0888330
\(262\) 4.64495 0.286966
\(263\) −23.6937 −1.46101 −0.730507 0.682905i \(-0.760716\pi\)
−0.730507 + 0.682905i \(0.760716\pi\)
\(264\) −2.66593 −0.164076
\(265\) −10.3553 −0.636119
\(266\) 4.35839 0.267230
\(267\) −15.1456 −0.926896
\(268\) −6.61521 −0.404088
\(269\) 25.6963 1.56673 0.783365 0.621561i \(-0.213501\pi\)
0.783365 + 0.621561i \(0.213501\pi\)
\(270\) 4.93840 0.300542
\(271\) 19.1460 1.16304 0.581519 0.813533i \(-0.302459\pi\)
0.581519 + 0.813533i \(0.302459\pi\)
\(272\) −3.53180 −0.214147
\(273\) 12.5254 0.758072
\(274\) 16.0268 0.968214
\(275\) 1.00000 0.0603023
\(276\) −1.10279 −0.0663801
\(277\) 12.6559 0.760421 0.380211 0.924900i \(-0.375851\pi\)
0.380211 + 0.924900i \(0.375851\pi\)
\(278\) −0.985838 −0.0591266
\(279\) 6.27586 0.375726
\(280\) −12.3325 −0.737006
\(281\) 7.90646 0.471660 0.235830 0.971794i \(-0.424219\pi\)
0.235830 + 0.971794i \(0.424219\pi\)
\(282\) −5.17680 −0.308274
\(283\) 11.8035 0.701647 0.350823 0.936442i \(-0.385902\pi\)
0.350823 + 0.936442i \(0.385902\pi\)
\(284\) −6.30768 −0.374292
\(285\) −0.870874 −0.0515862
\(286\) 3.86231 0.228383
\(287\) 8.51825 0.502817
\(288\) 9.71621 0.572533
\(289\) −12.4327 −0.731336
\(290\) 0.692643 0.0406734
\(291\) 5.46934 0.320618
\(292\) −2.89048 −0.169152
\(293\) 7.70878 0.450352 0.225176 0.974318i \(-0.427704\pi\)
0.225176 + 0.974318i \(0.427704\pi\)
\(294\) 8.69601 0.507162
\(295\) −10.5403 −0.613683
\(296\) 21.0658 1.22442
\(297\) 4.56476 0.264874
\(298\) 3.07071 0.177881
\(299\) 5.44944 0.315149
\(300\) 0.722469 0.0417118
\(301\) −45.3489 −2.61387
\(302\) 0.0526502 0.00302968
\(303\) −8.13659 −0.467435
\(304\) 1.65260 0.0947830
\(305\) −7.73205 −0.442736
\(306\) 5.18265 0.296273
\(307\) −0.978502 −0.0558460 −0.0279230 0.999610i \(-0.508889\pi\)
−0.0279230 + 0.999610i \(0.508889\pi\)
\(308\) −3.34212 −0.190435
\(309\) 12.6507 0.719674
\(310\) −3.02892 −0.172031
\(311\) −13.0195 −0.738268 −0.369134 0.929376i \(-0.620346\pi\)
−0.369134 + 0.929376i \(0.620346\pi\)
\(312\) 9.51759 0.538827
\(313\) −35.0042 −1.97856 −0.989278 0.146047i \(-0.953345\pi\)
−0.989278 + 0.146047i \(0.953345\pi\)
\(314\) −4.80276 −0.271035
\(315\) 9.03049 0.508811
\(316\) −4.56537 −0.256822
\(317\) 14.0969 0.791763 0.395882 0.918302i \(-0.370439\pi\)
0.395882 + 0.918302i \(0.370439\pi\)
\(318\) 9.75631 0.547107
\(319\) 0.640237 0.0358464
\(320\) −7.99454 −0.446908
\(321\) −9.95178 −0.555454
\(322\) 6.65272 0.370742
\(323\) −2.13712 −0.118913
\(324\) −2.28088 −0.126716
\(325\) −3.57009 −0.198033
\(326\) −21.0721 −1.16708
\(327\) −1.62307 −0.0897562
\(328\) 6.47270 0.357395
\(329\) −22.1357 −1.22038
\(330\) −0.942160 −0.0518642
\(331\) 2.32171 0.127613 0.0638065 0.997962i \(-0.479676\pi\)
0.0638065 + 0.997962i \(0.479676\pi\)
\(332\) −12.7953 −0.702231
\(333\) −15.4255 −0.845311
\(334\) −11.7406 −0.642419
\(335\) −7.97406 −0.435670
\(336\) 5.79803 0.316308
\(337\) 25.4277 1.38514 0.692568 0.721353i \(-0.256479\pi\)
0.692568 + 0.721353i \(0.256479\pi\)
\(338\) 0.275310 0.0149749
\(339\) 10.3539 0.562345
\(340\) 1.77294 0.0961510
\(341\) −2.79975 −0.151615
\(342\) −2.42506 −0.131132
\(343\) 8.98335 0.485055
\(344\) −34.4589 −1.85790
\(345\) −1.32932 −0.0715680
\(346\) 1.58584 0.0852553
\(347\) 0.658323 0.0353406 0.0176703 0.999844i \(-0.494375\pi\)
0.0176703 + 0.999844i \(0.494375\pi\)
\(348\) 0.462551 0.0247954
\(349\) −25.4110 −1.36022 −0.680110 0.733110i \(-0.738068\pi\)
−0.680110 + 0.733110i \(0.738068\pi\)
\(350\) −4.35839 −0.232966
\(351\) −16.2966 −0.869847
\(352\) −4.33454 −0.231032
\(353\) 18.3240 0.975287 0.487643 0.873043i \(-0.337857\pi\)
0.487643 + 0.873043i \(0.337857\pi\)
\(354\) 9.93069 0.527810
\(355\) −7.60336 −0.403545
\(356\) −14.4276 −0.764663
\(357\) −7.49795 −0.396834
\(358\) −19.0242 −1.00546
\(359\) −3.16020 −0.166789 −0.0833944 0.996517i \(-0.526576\pi\)
−0.0833944 + 0.996517i \(0.526576\pi\)
\(360\) 6.86193 0.361655
\(361\) 1.00000 0.0526316
\(362\) −1.18958 −0.0625229
\(363\) −0.870874 −0.0457091
\(364\) 11.9316 0.625388
\(365\) −3.48422 −0.182372
\(366\) 7.28482 0.380784
\(367\) −31.3432 −1.63610 −0.818050 0.575147i \(-0.804945\pi\)
−0.818050 + 0.575147i \(0.804945\pi\)
\(368\) 2.52255 0.131497
\(369\) −4.73966 −0.246737
\(370\) 7.44481 0.387037
\(371\) 41.7176 2.16587
\(372\) −2.02273 −0.104874
\(373\) 16.6133 0.860206 0.430103 0.902780i \(-0.358477\pi\)
0.430103 + 0.902780i \(0.358477\pi\)
\(374\) −2.31206 −0.119554
\(375\) 0.870874 0.0449718
\(376\) −16.8201 −0.867431
\(377\) −2.28570 −0.117720
\(378\) −19.8950 −1.02329
\(379\) −24.6701 −1.26722 −0.633610 0.773653i \(-0.718428\pi\)
−0.633610 + 0.773653i \(0.718428\pi\)
\(380\) −0.829591 −0.0425571
\(381\) 9.16119 0.469342
\(382\) −21.6708 −1.10878
\(383\) 34.0603 1.74040 0.870200 0.492699i \(-0.163990\pi\)
0.870200 + 0.492699i \(0.163990\pi\)
\(384\) −0.0175511 −0.000895651 0
\(385\) −4.02863 −0.205318
\(386\) 24.5645 1.25030
\(387\) 25.2326 1.28265
\(388\) 5.21007 0.264501
\(389\) 14.4879 0.734567 0.367284 0.930109i \(-0.380288\pi\)
0.367284 + 0.930109i \(0.380288\pi\)
\(390\) 3.36359 0.170322
\(391\) −3.26214 −0.164973
\(392\) 28.2545 1.42707
\(393\) 3.73911 0.188613
\(394\) 0.933312 0.0470196
\(395\) −5.50316 −0.276894
\(396\) 1.85959 0.0934480
\(397\) 14.7742 0.741496 0.370748 0.928734i \(-0.379101\pi\)
0.370748 + 0.928734i \(0.379101\pi\)
\(398\) −19.7749 −0.991227
\(399\) 3.50843 0.175641
\(400\) −1.65260 −0.0826299
\(401\) 0.975341 0.0487062 0.0243531 0.999703i \(-0.492247\pi\)
0.0243531 + 0.999703i \(0.492247\pi\)
\(402\) 7.51284 0.374706
\(403\) 9.99535 0.497904
\(404\) −7.75088 −0.385620
\(405\) −2.74940 −0.136619
\(406\) −2.79040 −0.138485
\(407\) 6.88152 0.341104
\(408\) −5.69741 −0.282064
\(409\) 11.6767 0.577377 0.288689 0.957423i \(-0.406781\pi\)
0.288689 + 0.957423i \(0.406781\pi\)
\(410\) 2.28750 0.112972
\(411\) 12.9013 0.636374
\(412\) 12.0510 0.593711
\(413\) 42.4632 2.08948
\(414\) −3.70165 −0.181926
\(415\) −15.4236 −0.757114
\(416\) 15.4747 0.758709
\(417\) −0.793583 −0.0388619
\(418\) 1.08185 0.0529152
\(419\) −9.20529 −0.449708 −0.224854 0.974392i \(-0.572190\pi\)
−0.224854 + 0.974392i \(0.572190\pi\)
\(420\) −2.91056 −0.142021
\(421\) −26.1447 −1.27421 −0.637106 0.770776i \(-0.719869\pi\)
−0.637106 + 0.770776i \(0.719869\pi\)
\(422\) −5.07567 −0.247080
\(423\) 12.3166 0.598853
\(424\) 31.6996 1.53947
\(425\) 2.13712 0.103666
\(426\) 7.16358 0.347077
\(427\) 31.1496 1.50743
\(428\) −9.48002 −0.458234
\(429\) 3.10910 0.150109
\(430\) −12.1781 −0.587278
\(431\) 17.5078 0.843320 0.421660 0.906754i \(-0.361448\pi\)
0.421660 + 0.906754i \(0.361448\pi\)
\(432\) −7.54371 −0.362947
\(433\) 23.2988 1.11967 0.559834 0.828605i \(-0.310865\pi\)
0.559834 + 0.828605i \(0.310865\pi\)
\(434\) 12.2024 0.585735
\(435\) 0.557566 0.0267332
\(436\) −1.54613 −0.0740463
\(437\) 1.52642 0.0730183
\(438\) 3.28269 0.156853
\(439\) 22.5592 1.07669 0.538345 0.842725i \(-0.319050\pi\)
0.538345 + 0.842725i \(0.319050\pi\)
\(440\) −3.06121 −0.145937
\(441\) −20.6895 −0.985213
\(442\) 8.25424 0.392614
\(443\) −2.78821 −0.132472 −0.0662360 0.997804i \(-0.521099\pi\)
−0.0662360 + 0.997804i \(0.521099\pi\)
\(444\) 4.97169 0.235946
\(445\) −17.3913 −0.824425
\(446\) −0.926132 −0.0438536
\(447\) 2.47187 0.116915
\(448\) 32.2071 1.52164
\(449\) −28.2226 −1.33191 −0.665953 0.745994i \(-0.731975\pi\)
−0.665953 + 0.745994i \(0.731975\pi\)
\(450\) 2.42506 0.114318
\(451\) 2.11443 0.0995645
\(452\) 9.86305 0.463919
\(453\) 0.0423825 0.00199130
\(454\) −4.90491 −0.230199
\(455\) 14.3826 0.674265
\(456\) 2.66593 0.124843
\(457\) −37.0632 −1.73374 −0.866872 0.498530i \(-0.833873\pi\)
−0.866872 + 0.498530i \(0.833873\pi\)
\(458\) 30.8404 1.44108
\(459\) 9.75544 0.455345
\(460\) −1.26630 −0.0590415
\(461\) 19.8588 0.924917 0.462459 0.886641i \(-0.346967\pi\)
0.462459 + 0.886641i \(0.346967\pi\)
\(462\) 3.79561 0.176588
\(463\) −14.9995 −0.697084 −0.348542 0.937293i \(-0.613323\pi\)
−0.348542 + 0.937293i \(0.613323\pi\)
\(464\) −1.05805 −0.0491189
\(465\) −2.43823 −0.113070
\(466\) −6.07271 −0.281313
\(467\) 34.4593 1.59459 0.797293 0.603592i \(-0.206264\pi\)
0.797293 + 0.603592i \(0.206264\pi\)
\(468\) −6.63890 −0.306883
\(469\) 32.1246 1.48337
\(470\) −5.94436 −0.274193
\(471\) −3.86614 −0.178142
\(472\) 32.2662 1.48517
\(473\) −11.2566 −0.517581
\(474\) 5.18485 0.238148
\(475\) −1.00000 −0.0458831
\(476\) −7.14251 −0.327376
\(477\) −23.2121 −1.06281
\(478\) 22.4150 1.02524
\(479\) 16.8146 0.768280 0.384140 0.923275i \(-0.374498\pi\)
0.384140 + 0.923275i \(0.374498\pi\)
\(480\) −3.77484 −0.172297
\(481\) −24.5676 −1.12019
\(482\) −12.3602 −0.562993
\(483\) 5.35533 0.243676
\(484\) −0.829591 −0.0377087
\(485\) 6.28029 0.285173
\(486\) 17.4056 0.789533
\(487\) 31.1763 1.41273 0.706366 0.707847i \(-0.250333\pi\)
0.706366 + 0.707847i \(0.250333\pi\)
\(488\) 23.6694 1.07146
\(489\) −16.9627 −0.767078
\(490\) 9.98538 0.451094
\(491\) 9.96801 0.449850 0.224925 0.974376i \(-0.427786\pi\)
0.224925 + 0.974376i \(0.427786\pi\)
\(492\) 1.52761 0.0688700
\(493\) 1.36826 0.0616235
\(494\) −3.86231 −0.173774
\(495\) 2.24158 0.100751
\(496\) 4.62686 0.207752
\(497\) 30.6311 1.37399
\(498\) 14.5315 0.651171
\(499\) 14.1464 0.633279 0.316640 0.948546i \(-0.397445\pi\)
0.316640 + 0.948546i \(0.397445\pi\)
\(500\) 0.829591 0.0371004
\(501\) −9.45101 −0.422240
\(502\) 11.2383 0.501590
\(503\) −22.9128 −1.02163 −0.510816 0.859690i \(-0.670657\pi\)
−0.510816 + 0.859690i \(0.670657\pi\)
\(504\) −27.6442 −1.23137
\(505\) −9.34301 −0.415759
\(506\) 1.65136 0.0734119
\(507\) 0.221620 0.00984248
\(508\) 8.72690 0.387194
\(509\) −32.1228 −1.42382 −0.711908 0.702273i \(-0.752169\pi\)
−0.711908 + 0.702273i \(0.752169\pi\)
\(510\) −2.01351 −0.0891597
\(511\) 14.0366 0.620945
\(512\) 17.2811 0.763726
\(513\) −4.56476 −0.201539
\(514\) −4.91109 −0.216619
\(515\) 14.5265 0.640112
\(516\) −8.13258 −0.358017
\(517\) −5.49461 −0.241653
\(518\) −29.9924 −1.31779
\(519\) 1.27657 0.0560354
\(520\) 10.9288 0.479258
\(521\) 22.6591 0.992713 0.496356 0.868119i \(-0.334671\pi\)
0.496356 + 0.868119i \(0.334671\pi\)
\(522\) 1.55261 0.0679560
\(523\) −1.68836 −0.0738269 −0.0369134 0.999318i \(-0.511753\pi\)
−0.0369134 + 0.999318i \(0.511753\pi\)
\(524\) 3.56186 0.155600
\(525\) −3.50843 −0.153121
\(526\) 25.6331 1.11766
\(527\) −5.98341 −0.260641
\(528\) 1.43921 0.0626334
\(529\) −20.6701 −0.898698
\(530\) 11.2029 0.486623
\(531\) −23.6270 −1.02533
\(532\) 3.34212 0.144899
\(533\) −7.54869 −0.326970
\(534\) 16.3853 0.709063
\(535\) −11.4273 −0.494047
\(536\) 24.4103 1.05436
\(537\) −15.3142 −0.660855
\(538\) −27.7997 −1.19853
\(539\) 9.22988 0.397559
\(540\) 3.78688 0.162961
\(541\) 0.0300570 0.00129225 0.000646125 1.00000i \(-0.499794\pi\)
0.000646125 1.00000i \(0.499794\pi\)
\(542\) −20.7132 −0.889708
\(543\) −0.957591 −0.0410942
\(544\) −9.26344 −0.397167
\(545\) −1.86373 −0.0798334
\(546\) −13.5507 −0.579915
\(547\) −18.4920 −0.790662 −0.395331 0.918539i \(-0.629370\pi\)
−0.395331 + 0.918539i \(0.629370\pi\)
\(548\) 12.2897 0.524990
\(549\) −17.3320 −0.739711
\(550\) −1.08185 −0.0461304
\(551\) −0.640237 −0.0272750
\(552\) 4.06931 0.173201
\(553\) 22.1702 0.942773
\(554\) −13.6919 −0.581712
\(555\) 5.99294 0.254386
\(556\) −0.755963 −0.0320600
\(557\) −40.9492 −1.73507 −0.867537 0.497372i \(-0.834298\pi\)
−0.867537 + 0.497372i \(0.834298\pi\)
\(558\) −6.78957 −0.287425
\(559\) 40.1872 1.69974
\(560\) 6.65771 0.281340
\(561\) −1.86117 −0.0785785
\(562\) −8.55364 −0.360814
\(563\) −8.92339 −0.376076 −0.188038 0.982162i \(-0.560213\pi\)
−0.188038 + 0.982162i \(0.560213\pi\)
\(564\) −3.96968 −0.167154
\(565\) 11.8891 0.500176
\(566\) −12.7697 −0.536750
\(567\) 11.0763 0.465162
\(568\) 23.2755 0.976616
\(569\) −47.1486 −1.97657 −0.988286 0.152613i \(-0.951231\pi\)
−0.988286 + 0.152613i \(0.951231\pi\)
\(570\) 0.942160 0.0394627
\(571\) 29.7980 1.24701 0.623504 0.781820i \(-0.285709\pi\)
0.623504 + 0.781820i \(0.285709\pi\)
\(572\) 2.96171 0.123835
\(573\) −17.4447 −0.728761
\(574\) −9.21551 −0.384648
\(575\) −1.52642 −0.0636559
\(576\) −17.9204 −0.746682
\(577\) −1.81737 −0.0756582 −0.0378291 0.999284i \(-0.512044\pi\)
−0.0378291 + 0.999284i \(0.512044\pi\)
\(578\) 13.4504 0.559462
\(579\) 19.7740 0.821781
\(580\) 0.531134 0.0220542
\(581\) 62.1359 2.57783
\(582\) −5.91703 −0.245269
\(583\) 10.3553 0.428871
\(584\) 10.6659 0.441359
\(585\) −8.00263 −0.330868
\(586\) −8.33978 −0.344513
\(587\) −17.9910 −0.742567 −0.371284 0.928519i \(-0.621082\pi\)
−0.371284 + 0.928519i \(0.621082\pi\)
\(588\) 6.66830 0.274996
\(589\) 2.79975 0.115362
\(590\) 11.4031 0.469459
\(591\) 0.751300 0.0309044
\(592\) −11.3724 −0.467402
\(593\) −28.7694 −1.18142 −0.590710 0.806884i \(-0.701152\pi\)
−0.590710 + 0.806884i \(0.701152\pi\)
\(594\) −4.93840 −0.202625
\(595\) −8.60968 −0.352962
\(596\) 2.35469 0.0964518
\(597\) −15.9185 −0.651500
\(598\) −5.89550 −0.241085
\(599\) −9.51516 −0.388779 −0.194389 0.980924i \(-0.562273\pi\)
−0.194389 + 0.980924i \(0.562273\pi\)
\(600\) −2.66593 −0.108836
\(601\) 1.39250 0.0568011 0.0284006 0.999597i \(-0.490959\pi\)
0.0284006 + 0.999597i \(0.490959\pi\)
\(602\) 49.0609 1.99957
\(603\) −17.8745 −0.727905
\(604\) 0.0403734 0.00164277
\(605\) −1.00000 −0.0406558
\(606\) 8.80261 0.357581
\(607\) −47.4420 −1.92561 −0.962805 0.270198i \(-0.912911\pi\)
−0.962805 + 0.270198i \(0.912911\pi\)
\(608\) 4.33454 0.175789
\(609\) −2.24623 −0.0910217
\(610\) 8.36495 0.338687
\(611\) 19.6162 0.793587
\(612\) 3.97418 0.160647
\(613\) −7.35538 −0.297081 −0.148540 0.988906i \(-0.547458\pi\)
−0.148540 + 0.988906i \(0.547458\pi\)
\(614\) 1.05860 0.0427215
\(615\) 1.84140 0.0742525
\(616\) 12.3325 0.496889
\(617\) −25.6134 −1.03116 −0.515578 0.856843i \(-0.672423\pi\)
−0.515578 + 0.856843i \(0.672423\pi\)
\(618\) −13.6862 −0.550541
\(619\) −15.6734 −0.629969 −0.314984 0.949097i \(-0.601999\pi\)
−0.314984 + 0.949097i \(0.601999\pi\)
\(620\) −2.32265 −0.0932797
\(621\) −6.96771 −0.279605
\(622\) 14.0852 0.564765
\(623\) 70.0630 2.80702
\(624\) −5.13809 −0.205688
\(625\) 1.00000 0.0400000
\(626\) 37.8695 1.51357
\(627\) 0.870874 0.0347794
\(628\) −3.68286 −0.146962
\(629\) 14.7067 0.586393
\(630\) −9.76968 −0.389233
\(631\) −11.7074 −0.466065 −0.233032 0.972469i \(-0.574865\pi\)
−0.233032 + 0.972469i \(0.574865\pi\)
\(632\) 16.8463 0.670110
\(633\) −4.08583 −0.162397
\(634\) −15.2508 −0.605689
\(635\) 10.5195 0.417455
\(636\) 7.48136 0.296655
\(637\) −32.9515 −1.30558
\(638\) −0.692643 −0.0274220
\(639\) −17.0435 −0.674231
\(640\) −0.0201534 −0.000796634 0
\(641\) 22.6311 0.893874 0.446937 0.894566i \(-0.352515\pi\)
0.446937 + 0.894566i \(0.352515\pi\)
\(642\) 10.7664 0.424915
\(643\) −50.5556 −1.99372 −0.996860 0.0791848i \(-0.974768\pi\)
−0.996860 + 0.0791848i \(0.974768\pi\)
\(644\) 5.10146 0.201026
\(645\) −9.80313 −0.385998
\(646\) 2.31206 0.0909666
\(647\) −29.2983 −1.15183 −0.575917 0.817508i \(-0.695355\pi\)
−0.575917 + 0.817508i \(0.695355\pi\)
\(648\) 8.41649 0.330631
\(649\) 10.5403 0.413745
\(650\) 3.86231 0.151492
\(651\) 9.82274 0.384983
\(652\) −16.1586 −0.632818
\(653\) −4.84850 −0.189736 −0.0948682 0.995490i \(-0.530243\pi\)
−0.0948682 + 0.995490i \(0.530243\pi\)
\(654\) 1.75593 0.0686623
\(655\) 4.29351 0.167761
\(656\) −3.49430 −0.136429
\(657\) −7.81015 −0.304703
\(658\) 23.9477 0.933577
\(659\) 15.7032 0.611708 0.305854 0.952078i \(-0.401058\pi\)
0.305854 + 0.952078i \(0.401058\pi\)
\(660\) −0.722469 −0.0281221
\(661\) −13.3369 −0.518744 −0.259372 0.965777i \(-0.583516\pi\)
−0.259372 + 0.965777i \(0.583516\pi\)
\(662\) −2.51176 −0.0976222
\(663\) 6.64452 0.258052
\(664\) 47.2148 1.83229
\(665\) 4.02863 0.156224
\(666\) 16.6881 0.646651
\(667\) −0.977267 −0.0378399
\(668\) −9.00299 −0.348336
\(669\) −0.745520 −0.0288235
\(670\) 8.62678 0.333282
\(671\) 7.73205 0.298492
\(672\) 15.2074 0.586640
\(673\) 44.6837 1.72243 0.861214 0.508242i \(-0.169704\pi\)
0.861214 + 0.508242i \(0.169704\pi\)
\(674\) −27.5091 −1.05961
\(675\) 4.56476 0.175698
\(676\) 0.211114 0.00811976
\(677\) 11.2362 0.431843 0.215922 0.976411i \(-0.430724\pi\)
0.215922 + 0.976411i \(0.430724\pi\)
\(678\) −11.2014 −0.430187
\(679\) −25.3010 −0.970962
\(680\) −6.54217 −0.250881
\(681\) −3.94837 −0.151302
\(682\) 3.02892 0.115983
\(683\) −40.5671 −1.55226 −0.776128 0.630575i \(-0.782819\pi\)
−0.776128 + 0.630575i \(0.782819\pi\)
\(684\) −1.85959 −0.0711033
\(685\) 14.8142 0.566021
\(686\) −9.71867 −0.371061
\(687\) 24.8260 0.947170
\(688\) 18.6027 0.709222
\(689\) −36.9692 −1.40841
\(690\) 1.43813 0.0547485
\(691\) 9.10082 0.346212 0.173106 0.984903i \(-0.444620\pi\)
0.173106 + 0.984903i \(0.444620\pi\)
\(692\) 1.21606 0.0462276
\(693\) −9.03049 −0.343040
\(694\) −0.712209 −0.0270351
\(695\) −0.911248 −0.0345656
\(696\) −1.70682 −0.0646970
\(697\) 4.51879 0.171161
\(698\) 27.4910 1.04055
\(699\) −4.88843 −0.184897
\(700\) −3.34212 −0.126320
\(701\) 24.5747 0.928175 0.464088 0.885789i \(-0.346382\pi\)
0.464088 + 0.885789i \(0.346382\pi\)
\(702\) 17.6305 0.665421
\(703\) −6.88152 −0.259542
\(704\) 7.99454 0.301306
\(705\) −4.78511 −0.180218
\(706\) −19.8239 −0.746081
\(707\) 37.6396 1.41558
\(708\) 7.61508 0.286192
\(709\) −6.77169 −0.254316 −0.127158 0.991882i \(-0.540586\pi\)
−0.127158 + 0.991882i \(0.540586\pi\)
\(710\) 8.22573 0.308706
\(711\) −12.3358 −0.462627
\(712\) 53.2382 1.99519
\(713\) 4.27358 0.160047
\(714\) 8.11169 0.303572
\(715\) 3.57009 0.133514
\(716\) −14.5882 −0.545186
\(717\) 18.0437 0.673853
\(718\) 3.41887 0.127591
\(719\) 36.1548 1.34835 0.674173 0.738574i \(-0.264500\pi\)
0.674173 + 0.738574i \(0.264500\pi\)
\(720\) −3.70443 −0.138056
\(721\) −58.5217 −2.17946
\(722\) −1.08185 −0.0402625
\(723\) −9.94978 −0.370036
\(724\) −0.912197 −0.0339015
\(725\) 0.640237 0.0237778
\(726\) 0.942160 0.0349668
\(727\) 10.4462 0.387427 0.193714 0.981058i \(-0.437947\pi\)
0.193714 + 0.981058i \(0.437947\pi\)
\(728\) −44.0280 −1.63179
\(729\) 5.76299 0.213444
\(730\) 3.76942 0.139513
\(731\) −24.0568 −0.889774
\(732\) 5.58617 0.206471
\(733\) −0.656143 −0.0242352 −0.0121176 0.999927i \(-0.503857\pi\)
−0.0121176 + 0.999927i \(0.503857\pi\)
\(734\) 33.9087 1.25159
\(735\) 8.03806 0.296488
\(736\) 6.61631 0.243880
\(737\) 7.97406 0.293728
\(738\) 5.12762 0.188750
\(739\) −42.0114 −1.54541 −0.772707 0.634762i \(-0.781098\pi\)
−0.772707 + 0.634762i \(0.781098\pi\)
\(740\) 5.70885 0.209861
\(741\) −3.10910 −0.114216
\(742\) −45.1323 −1.65686
\(743\) −6.16676 −0.226236 −0.113118 0.993582i \(-0.536084\pi\)
−0.113118 + 0.993582i \(0.536084\pi\)
\(744\) 7.46393 0.273641
\(745\) 2.83838 0.103990
\(746\) −17.9732 −0.658046
\(747\) −34.5732 −1.26497
\(748\) −1.77294 −0.0648250
\(749\) 46.0366 1.68214
\(750\) −0.942160 −0.0344028
\(751\) 37.9420 1.38452 0.692262 0.721646i \(-0.256614\pi\)
0.692262 + 0.721646i \(0.256614\pi\)
\(752\) 9.08037 0.331127
\(753\) 9.04665 0.329678
\(754\) 2.47280 0.0900539
\(755\) 0.0486666 0.00177116
\(756\) −15.2559 −0.554853
\(757\) −18.3569 −0.667192 −0.333596 0.942716i \(-0.608262\pi\)
−0.333596 + 0.942716i \(0.608262\pi\)
\(758\) 26.6895 0.969407
\(759\) 1.32932 0.0482511
\(760\) 3.06121 0.111042
\(761\) 18.0726 0.655130 0.327565 0.944829i \(-0.393772\pi\)
0.327565 + 0.944829i \(0.393772\pi\)
\(762\) −9.91107 −0.359040
\(763\) 7.50828 0.271818
\(764\) −16.6177 −0.601207
\(765\) 4.79053 0.173202
\(766\) −36.8483 −1.33138
\(767\) −37.6300 −1.35874
\(768\) 13.9435 0.503142
\(769\) 35.1639 1.26804 0.634021 0.773316i \(-0.281403\pi\)
0.634021 + 0.773316i \(0.281403\pi\)
\(770\) 4.35839 0.157066
\(771\) −3.95334 −0.142376
\(772\) 18.8366 0.677946
\(773\) 15.9468 0.573568 0.286784 0.957995i \(-0.407414\pi\)
0.286784 + 0.957995i \(0.407414\pi\)
\(774\) −27.2981 −0.981208
\(775\) −2.79975 −0.100570
\(776\) −19.2252 −0.690146
\(777\) −24.1434 −0.866138
\(778\) −15.6738 −0.561934
\(779\) −2.11443 −0.0757573
\(780\) 2.57928 0.0923530
\(781\) 7.60336 0.272070
\(782\) 3.52916 0.126202
\(783\) 2.92252 0.104443
\(784\) −15.2533 −0.544760
\(785\) −4.43937 −0.158448
\(786\) −4.04517 −0.144287
\(787\) 52.0777 1.85637 0.928185 0.372119i \(-0.121369\pi\)
0.928185 + 0.372119i \(0.121369\pi\)
\(788\) 0.715685 0.0254952
\(789\) 20.6342 0.734598
\(790\) 5.95362 0.211820
\(791\) −47.8966 −1.70301
\(792\) −6.86193 −0.243828
\(793\) −27.6041 −0.980250
\(794\) −15.9835 −0.567234
\(795\) 9.01814 0.319840
\(796\) −15.1639 −0.537469
\(797\) −41.6490 −1.47528 −0.737641 0.675193i \(-0.764060\pi\)
−0.737641 + 0.675193i \(0.764060\pi\)
\(798\) −3.79561 −0.134363
\(799\) −11.7426 −0.415425
\(800\) −4.33454 −0.153249
\(801\) −38.9839 −1.37743
\(802\) −1.05518 −0.0372596
\(803\) 3.48422 0.122955
\(804\) 5.76102 0.203175
\(805\) 6.14937 0.216737
\(806\) −10.8135 −0.380890
\(807\) −22.3783 −0.787752
\(808\) 28.6009 1.00618
\(809\) 32.6440 1.14770 0.573851 0.818960i \(-0.305449\pi\)
0.573851 + 0.818960i \(0.305449\pi\)
\(810\) 2.97445 0.104512
\(811\) −7.82261 −0.274689 −0.137345 0.990523i \(-0.543857\pi\)
−0.137345 + 0.990523i \(0.543857\pi\)
\(812\) −2.13975 −0.0750903
\(813\) −16.6738 −0.584775
\(814\) −7.44481 −0.260940
\(815\) −19.4777 −0.682276
\(816\) 3.07576 0.107673
\(817\) 11.2566 0.393820
\(818\) −12.6325 −0.441686
\(819\) 32.2396 1.12654
\(820\) 1.75411 0.0612562
\(821\) 38.0387 1.32756 0.663780 0.747927i \(-0.268951\pi\)
0.663780 + 0.747927i \(0.268951\pi\)
\(822\) −13.9573 −0.486818
\(823\) 39.7091 1.38417 0.692087 0.721815i \(-0.256692\pi\)
0.692087 + 0.721815i \(0.256692\pi\)
\(824\) −44.4685 −1.54913
\(825\) −0.870874 −0.0303200
\(826\) −45.9390 −1.59842
\(827\) −38.3592 −1.33388 −0.666940 0.745111i \(-0.732396\pi\)
−0.666940 + 0.745111i \(0.732396\pi\)
\(828\) −2.83851 −0.0986450
\(829\) 3.48941 0.121192 0.0605960 0.998162i \(-0.480700\pi\)
0.0605960 + 0.998162i \(0.480700\pi\)
\(830\) 16.6861 0.579182
\(831\) −11.0217 −0.382340
\(832\) −28.5412 −0.989488
\(833\) 19.7254 0.683444
\(834\) 0.858541 0.0297288
\(835\) −10.8523 −0.375560
\(836\) 0.829591 0.0286920
\(837\) −12.7802 −0.441748
\(838\) 9.95878 0.344020
\(839\) 39.6707 1.36959 0.684793 0.728738i \(-0.259893\pi\)
0.684793 + 0.728738i \(0.259893\pi\)
\(840\) 10.7400 0.370566
\(841\) −28.5901 −0.985865
\(842\) 28.2847 0.974756
\(843\) −6.88553 −0.237150
\(844\) −3.89214 −0.133973
\(845\) 0.254480 0.00875436
\(846\) −13.3248 −0.458115
\(847\) 4.02863 0.138425
\(848\) −17.1131 −0.587666
\(849\) −10.2794 −0.352788
\(850\) −2.31206 −0.0793029
\(851\) −10.5041 −0.360075
\(852\) 5.49320 0.188194
\(853\) 9.19215 0.314733 0.157367 0.987540i \(-0.449700\pi\)
0.157367 + 0.987540i \(0.449700\pi\)
\(854\) −33.6993 −1.15317
\(855\) −2.24158 −0.0766603
\(856\) 34.9814 1.19564
\(857\) −40.0597 −1.36841 −0.684207 0.729288i \(-0.739851\pi\)
−0.684207 + 0.729288i \(0.739851\pi\)
\(858\) −3.36359 −0.114831
\(859\) 13.2980 0.453722 0.226861 0.973927i \(-0.427154\pi\)
0.226861 + 0.973927i \(0.427154\pi\)
\(860\) −9.33841 −0.318437
\(861\) −7.41833 −0.252816
\(862\) −18.9409 −0.645129
\(863\) −35.6269 −1.21275 −0.606377 0.795177i \(-0.707378\pi\)
−0.606377 + 0.795177i \(0.707378\pi\)
\(864\) −19.7861 −0.673138
\(865\) 1.46585 0.0498405
\(866\) −25.2059 −0.856530
\(867\) 10.8273 0.367715
\(868\) 9.35709 0.317600
\(869\) 5.50316 0.186682
\(870\) −0.603205 −0.0204506
\(871\) −28.4681 −0.964605
\(872\) 5.70526 0.193204
\(873\) 14.0777 0.476460
\(874\) −1.65136 −0.0558581
\(875\) −4.02863 −0.136193
\(876\) 2.51724 0.0850498
\(877\) −36.2257 −1.22325 −0.611627 0.791146i \(-0.709485\pi\)
−0.611627 + 0.791146i \(0.709485\pi\)
\(878\) −24.4057 −0.823654
\(879\) −6.71338 −0.226437
\(880\) 1.65260 0.0557091
\(881\) −5.12526 −0.172674 −0.0863372 0.996266i \(-0.527516\pi\)
−0.0863372 + 0.996266i \(0.527516\pi\)
\(882\) 22.3830 0.753675
\(883\) 38.1184 1.28279 0.641394 0.767212i \(-0.278357\pi\)
0.641394 + 0.767212i \(0.278357\pi\)
\(884\) 6.32954 0.212885
\(885\) 9.17932 0.308560
\(886\) 3.01644 0.101339
\(887\) 31.1569 1.04615 0.523074 0.852287i \(-0.324785\pi\)
0.523074 + 0.852287i \(0.324785\pi\)
\(888\) −18.3456 −0.615639
\(889\) −42.3793 −1.42136
\(890\) 18.8148 0.630674
\(891\) 2.74940 0.0921085
\(892\) −0.710179 −0.0237786
\(893\) 5.49461 0.183870
\(894\) −2.67420 −0.0894387
\(895\) −17.5848 −0.587795
\(896\) 0.0811907 0.00271239
\(897\) −4.74577 −0.158457
\(898\) 30.5327 1.01889
\(899\) −1.79250 −0.0597833
\(900\) 1.85959 0.0619864
\(901\) 22.1305 0.737273
\(902\) −2.28750 −0.0761655
\(903\) 39.4932 1.31425
\(904\) −36.3948 −1.21047
\(905\) −1.09957 −0.0365511
\(906\) −0.0458517 −0.00152332
\(907\) −23.4382 −0.778253 −0.389126 0.921184i \(-0.627223\pi\)
−0.389126 + 0.921184i \(0.627223\pi\)
\(908\) −3.76120 −0.124820
\(909\) −20.9431 −0.694638
\(910\) −15.5598 −0.515804
\(911\) −32.7985 −1.08666 −0.543331 0.839519i \(-0.682837\pi\)
−0.543331 + 0.839519i \(0.682837\pi\)
\(912\) −1.43921 −0.0476568
\(913\) 15.4236 0.510446
\(914\) 40.0970 1.32629
\(915\) 6.73364 0.222607
\(916\) 23.6491 0.781388
\(917\) −17.2970 −0.571196
\(918\) −10.5540 −0.348333
\(919\) 11.6985 0.385898 0.192949 0.981209i \(-0.438195\pi\)
0.192949 + 0.981209i \(0.438195\pi\)
\(920\) 4.67267 0.154053
\(921\) 0.852152 0.0280794
\(922\) −21.4843 −0.707549
\(923\) −27.1447 −0.893477
\(924\) 2.91056 0.0957505
\(925\) 6.88152 0.226263
\(926\) 16.2272 0.533260
\(927\) 32.5622 1.06948
\(928\) −2.77513 −0.0910982
\(929\) −4.68472 −0.153701 −0.0768503 0.997043i \(-0.524486\pi\)
−0.0768503 + 0.997043i \(0.524486\pi\)
\(930\) 2.63781 0.0864972
\(931\) −9.22988 −0.302497
\(932\) −4.65669 −0.152535
\(933\) 11.3383 0.371201
\(934\) −37.2800 −1.21984
\(935\) −2.13712 −0.0698914
\(936\) 24.4977 0.800732
\(937\) 2.22853 0.0728029 0.0364015 0.999337i \(-0.488410\pi\)
0.0364015 + 0.999337i \(0.488410\pi\)
\(938\) −34.7541 −1.13476
\(939\) 30.4843 0.994817
\(940\) −4.55827 −0.148675
\(941\) 22.5997 0.736730 0.368365 0.929681i \(-0.379918\pi\)
0.368365 + 0.929681i \(0.379918\pi\)
\(942\) 4.18260 0.136276
\(943\) −3.22750 −0.105102
\(944\) −17.4190 −0.566939
\(945\) −18.3897 −0.598218
\(946\) 12.1781 0.395943
\(947\) 49.0872 1.59512 0.797560 0.603239i \(-0.206124\pi\)
0.797560 + 0.603239i \(0.206124\pi\)
\(948\) 3.97586 0.129130
\(949\) −12.4390 −0.403786
\(950\) 1.08185 0.0351000
\(951\) −12.2767 −0.398098
\(952\) 26.3560 0.854203
\(953\) 1.08440 0.0351272 0.0175636 0.999846i \(-0.494409\pi\)
0.0175636 + 0.999846i \(0.494409\pi\)
\(954\) 25.1122 0.813036
\(955\) −20.0312 −0.648194
\(956\) 17.1883 0.555910
\(957\) −0.557566 −0.0180235
\(958\) −18.1910 −0.587724
\(959\) −59.6809 −1.92720
\(960\) 6.96224 0.224705
\(961\) −23.1614 −0.747142
\(962\) 26.5786 0.856929
\(963\) −25.6153 −0.825441
\(964\) −9.47811 −0.305269
\(965\) 22.7060 0.730931
\(966\) −5.79368 −0.186409
\(967\) −51.3313 −1.65070 −0.825351 0.564620i \(-0.809023\pi\)
−0.825351 + 0.564620i \(0.809023\pi\)
\(968\) 3.06121 0.0983909
\(969\) 1.86117 0.0597893
\(970\) −6.79436 −0.218154
\(971\) −22.9688 −0.737104 −0.368552 0.929607i \(-0.620146\pi\)
−0.368552 + 0.929607i \(0.620146\pi\)
\(972\) 13.3470 0.428105
\(973\) 3.67108 0.117690
\(974\) −33.7282 −1.08072
\(975\) 3.10910 0.0995708
\(976\) −12.7780 −0.409013
\(977\) −21.2040 −0.678376 −0.339188 0.940719i \(-0.610152\pi\)
−0.339188 + 0.940719i \(0.610152\pi\)
\(978\) 18.3511 0.586805
\(979\) 17.3913 0.555827
\(980\) 7.65702 0.244594
\(981\) −4.17769 −0.133383
\(982\) −10.7839 −0.344129
\(983\) 5.95713 0.190003 0.0950014 0.995477i \(-0.469714\pi\)
0.0950014 + 0.995477i \(0.469714\pi\)
\(984\) −5.63691 −0.179698
\(985\) 0.862697 0.0274878
\(986\) −1.48026 −0.0471412
\(987\) 19.2775 0.613608
\(988\) −2.96171 −0.0942246
\(989\) 17.1823 0.546366
\(990\) −2.42506 −0.0770735
\(991\) −4.63897 −0.147362 −0.0736809 0.997282i \(-0.523475\pi\)
−0.0736809 + 0.997282i \(0.523475\pi\)
\(992\) 12.1356 0.385307
\(993\) −2.02192 −0.0641638
\(994\) −33.1384 −1.05109
\(995\) −18.2787 −0.579475
\(996\) 11.1431 0.353082
\(997\) −30.8647 −0.977495 −0.488747 0.872425i \(-0.662546\pi\)
−0.488747 + 0.872425i \(0.662546\pi\)
\(998\) −15.3043 −0.484450
\(999\) 31.4125 0.993847
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 1045.2.a.h.1.3 7
3.2 odd 2 9405.2.a.bd.1.5 7
5.4 even 2 5225.2.a.m.1.5 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.h.1.3 7 1.1 even 1 trivial
5225.2.a.m.1.5 7 5.4 even 2
9405.2.a.bd.1.5 7 3.2 odd 2